Improving Applied Mathematics Education (SEMA SIMAI Springer Series, 7) 3030617165, 9783030617165

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ICIAM 2019 SEMA SIMAI Springer Series 7

Ron Buckmire Jessica M. Libertini Eds.

Improving Applied Mathematics Education

SEMA SIMAI Springer Series ICIAM 2019 SEMA SIMAI Springer Series Volume 7

Editor-in-Chief Amadeu Delshams, Departament de Matemàtiques and Laboratory of Geometry and Dynamical Systems, Universitat Politècnica de Catalunya, Barcelona, Spain Series Editors Francesc Arandiga Llaudes, Departamento de Matemàtica Aplicada, Universitat de València, Valencia, Spain Macarena Gómez Mármol, Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Sevilla, Spain Francisco M. Guillén-González, Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Sevilla, Spain Francisco Ortegón Gallego, Departamento de Matemáticas, Facultad de Ciencias del Mar y Ambientales, Universidad de Cádiz, Puerto Real, Spain Carlos Parés Madroñal, Departamento Análisis Matemático, Estadística e I.O., Matemática Aplicada, Universidad de Málaga, Málaga, Spain Peregrina Quintela, Department of Applied Mathematics, Faculty of Mathematics, Universidade de Santiago de Compostela, Santiago de Compostela, Spain Carlos Vázquez-Cendón, Department of Mathematics, Faculty of Informatics, Universidade da Coruña, A Coruña, Spain Sebastià Xambó-Descamps, Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

This sub-series of the SEMA SIMAI Springer Series aims to publish some of the most relevant results presented at the ICIAM 2019 conference held in Valencia in July 2019. The sub-series is managed by an independent Editorial Board, and will include peer-reviewed content only, including the Invited Speakers volume as well as books resulting from mini-symposia and collateral workshops. The series is aimed at providing useful reference material to academic and researchers at an international level.

More information about this subseries at http://www.springer.com/series/16499

Ron Buckmire • Jessica M. Libertini Editors

Improving Applied Mathematics Education

Editors Ron Buckmire Mathematics Department Occidental College Los Angeles, CA, USA

Jessica M. Libertini Geneva Centre for Security Policy (GCSP) Geneva, Switzerland Global Studies Institute University of Geneva Geneva, Switzerland

ISSN 2199-3041 ISSN 2199-305X (electronic) SEMA SIMAI Springer Series ISSN 2662-7183 ISSN 2662-7191 (electronic) ICIAM 2019 SEMA SIMAI Springer Series ISBN 978-3-030-61716-5 ISBN 978-3-030-61717-2 (eBook) https://doi.org/10.1007/978-3-030-61717-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

We are currently preparing students for jobs that don’t yet exist, using technologies that haven’t been invented, in order to solve problems we don’t even know are problems yet. Richard Riley, former US Secretary of Education Why do mathematics? I would say: mathematics helps people flourish. Mathematics is for human flourishing. Francis Su, past President of the Mathematical Association of America As mathematicians, we want to quantify everything: uncertainty and predictability, fairness and symmetry, complexity and simplicity, categories and diversity. We talk about numbers, improving the statistics, moving the needle. We want to reform, support, persist. Ami Radunskaya, past President of the Association for Women in Mathematics.

Preface

We are in a golden age of improving quality in undergraduate mathematics education. Innovators are sharing their successes and challenges in the classroom at conferences and in journals, and researchers are helping formalize the findings and measure the results. There is a shift toward the adoption of evidence-based and student-centered teaching practices. Faculty are engaging students in metacognition exercises, raising self-awareness, and developing independent learners. From mathematics for the liberal arts to differential equations, mathematics courses at all levels are motivating learners and improving problem-solving skills through the use of mathematical modeling of real-world, open-ended problems. And, this trend extends into the extracurricular space of modeling competitions. Furthermore, there is a renewed emphasis on humanity in mathematics—both in terms of seeing mathematics as a field that can help improve the human condition and in terms of seeing mathematicians as a collection of humans whose diversity and demographics should be representative of the general population. Mathematicians are engaging in policy and teaching courses that have students making a positive impact in their local communities and on the world at large. Many of the aforementioned evidence-based classroom practices have been proven to empower women, persons of color, and first-generation students, and instructors are increasingly aware of the importance of using equitable practices in their teaching. The 9th International Congress on Industrial and Applied Mathematics in Valencia, Spain (ICIAM 2019) held several sessions that focused on applied mathematics education, and this volume features some of the authors who presented at these sessions. The topics in this book span the issues discussed above: from the focus on the humanity of mathematicians to the use of student-centered practices to the use of competitions to create authentic problem-solving experiences for students. We hope that you enjoy this curated set of topics, and we look forward to ongoing conversations about them at future SIAM and ICIAM events. Los Angeles, CA, USA Geneva, Switzerland April 2020

Ron Buckmire Jessica M. Libertini

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Acknowledgments

The editors would like to acknowledge the Society for Industrial and Applied Mathematics (SIAM) and, in particular, the SIAM Activity Group on Applied Mathematics Education (SIAG-ED) for the inspiration and support of this collection.

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Contents

“Who Does the Math?”: On the Diversity and Demographics of the Mathematics Community in the USA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ron Buckmire

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Teaching Calculus as a Tool in the Twenty-First Century . . . . . . . . . . . . . . . . . . . . 13 Emma Smith Zbarsky, Gary Simundza, and Mel Henriksen Engaging Students in Applied Mathematics Education and Research for Global Problem Solving. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Wei Wang and Padmanabhan Seshaiyer Developing Non-Calculus Service Courses That Showcase the Applicability of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Lucas Castle Promoting Interdisciplinary and Mathematical Modelling Through Competitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Sergey Kushnarev and Jessica M. Libertini

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Editors and Contributors

About the Editors Ron Buckmire is a Professor of Mathematics at Occidental College in Los Angeles. He holds bachelor’s, master’s, and doctorate degrees in Mathematics from the Rensselaer Polytechnic Institute. He served for over four years as a Program Director in the Division of Undergraduate Education at the US National Science Foundation. He has published peer-reviewed work in a wide variety of research areas such as numerical analysis, mathematical modeling, machine learning, and mathematics education. Jessica M. Libertini received her Ph.D. in Applied Mathematics from Brown University. She has held faculty positions at the US Military Academy at West Point, University of Rhode Island, and Virginia Military Institute. In addition to her academic positions, she spent nearly a decade as an engineer and analyst with General Dynamics, and she served as a Science & Technology Policy Fellow in the Office of the Secretary of Defense. Her current work spans interdisciplinary education and international security.

Contributors Ron Buckmire Occidental College, Los Angeles, CA, USA Lucas Castle North Carolina State University, Raleigh, NC, USA Mel Henriksen Wentworth Institute of Technology, Boston, MA, USA Sergey Kushnarev Singapore University of Technology and Design, Singapore, Singapore

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Editors and Contributors

Jessica M. Libertini Geneva Centre for Security Policy (GCSP), Switzerland

Geneva,

Global Studies Institute, University of Geneva, Geneva, Switzerland Padmanbhan Seshaiyer George Mason University, Fairfax, VA, USA Gary Simundza Wentworth Institute of Technology, Boston, MA, USA Emma Smith Zbarsky Wentworth Institute of Technology, Boston, MA, USA

“Who Does the Math?”: On the Diversity and Demographics of the Mathematics Community in the USA Ron Buckmire

Abstract Mathematics is a human endeavor. In other words, mathematics is done, taught, discovered, and learned by people. Of course, people have various identifying characteristics and experiences that affect how they interact with other people and how people interact with them. In this paper I will be arguing that the identities of the people who are perceived as belonging to the mathematics community in the USA are important. I will present data about the diversity and demographics of the mathematics community in the USA and discuss the significance and implications of the underrepresentation of certain groups.

1 The Importance of STEM and Mathematics in the USA The role of mathematics and mathematics education in society has become increasingly prominent in recent times [1]. As the pace of technological change in society has increased, more people have realized the importance of quantitative skills in the workforce [2]. Terms like “STEM,” “data science,” “quantitative reasoning,” and “machine learning” have become more and more prevalent in and familiar to larger swaths of society. Mathematics is the “M” in the now-ubiquitous acronym “STEM” (science, technology, engineering, and mathematics). The importance of STEM has been recognized by various groups and organizations, such as the White House [3, 4], the National Research Council [5], the National Academy of Sciences [6], the National Science Teachers Association [7], and many others. As STEM has become more

R. Buckmire () Occidental College, Los Angeles, CA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 R. Buckmire, J. M. Libertini (eds.), Improving Applied Mathematics Education, SEMA SIMAI Springer Series 7, https://doi.org/10.1007/978-3-030-61717-2_1

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important the focus on STEM education as a means of enhancing “STEM literacy” [7] in the population has also increased. For example, in [8] the authors state: Over the last decade, numerous reports from U.S. business and government organizations have warned that the United States’ competitive edge in the global economy is eroding. These reports, along with a series of bills introduced in Congress and in state legislatures, call for an extensive effort to reform K–12 STEM education, and cultivate the next generation of skilled scientists, engineers, technicians, and science and mathematics educators.

One way of measuring the importance a society places on an issue is the assignment of resources to it. In the USA, the most prominent entity responsible for STEM education is the U.S. National Science Foundation (NSF). The NSF is an independent federal agency with an annual budget in the range of 8 billion US dollars and whose mission is “to promote the progress of science” [9]. There is an entire section of the NSF called the Directorate for Education and Human Resources (EHR) whose mission is to “achieve excellence” in STEM education [10]. EHR has an annual budget which has been in the range of 800–900 million US dollars for the last decade [11]. These numbers are some indication of the importance with which the USA treats STEM and STEM education, and by extension mathematics and mathematics education. The primary goal of this chapter is to convince the reader that “who does the math” is important. In other words, my thesis is that the (lack of) diversity in and demographics of the mathematics community deserve urgent attention. My argument proceeds as follows. Since STEM is an increasingly important feature of modern society and since mathematics is an important aspect of STEM, mathematics itself is important. In addition, mathematics is a human activity [12] and people have an infinite diversity of identifying characteristics. Thus it is important that the demographics of the people who do mathematics are representative of the community in which they live and contribute; in other words, the mathematics community should reflect the diversity of the society in which we live. One of the motivations for writing this chapter is the experiences of the author as a longtime member of the US mathematics community. These experiences encompass multiple ways one can participate in the mathematics community in the USA. One experience is as a professor of mathematics at a small, primarily undergraduate, liberal arts college in the USA for over 20 years. Another experience is as a graduate student and undergraduate student majoring in mathematics at Rensselaer Polytechnic Institute in Troy, New York (from 1986 to 1994). Yet another experience is as a government employee whose job required knowledge of mathematics and the mathematics community. This last experience comes from having served twice at the National Science Foundation (August 2011–August 2013 and May 2016–August 2018), both times working as a Program Director in the Division of Undergraduate Education, which is in the EHR Directorate. As an openly gay, African-American man who was not born in the USA, I have a great deal of experience being a member of an underrepresented group in the mathematics

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community. Thus, the importance of knowing “who does the math,” the realities of the underrepresentation of many groups in the mathematics community (due to race, ethnicity, gender, language, et cetera) and its concomitant implications are tangible and significant to me, and to many others who share identities that cause them to be visible minorities in most spaces associated with the mathematics community. This chapter is primarily intended to provide information about the demographics and (lack of) diversity in the mathematics community of the USA. A second(ary) intention is that highlighting this issue will lead to improvements in this area in the future. This chapter is a distillation and expansion of the thoughts and ideas presented during a talk given in a symposium on applied mathematics education which occurred at the 2019 International Congress of Industrial and Applied Mathematicians in Valencia, Spain. The rest of the paper will be organized as follows. Section 2 will discuss some proposed definitions of the term “mathematics community” (in the context of the USA) and provide information about one definition of the term. Section 3 will present and analyze some demographic data on the race and gender of undergraduate mathematics majors, undergraduate mathematics degree recipients, and mathematics Ph.D. recipients in the USA. The chapter will conclude with a short discussion on the various responses segments of the US mathematics community have had to the demographics and diversity of the community presented in earlier sections that confirm my thesis that knowing and discussing “who does the math” is important.

2 What is the “Mathematics Community”? In order to make the case that it is important that the mathematics community resemble the larger community from which it draws members, first we must try to come up with a reasonable definition of what the “mathematics community” is. Below are some examples of different ways one could define members of the mathematics community. 1. The set of individuals who are defined to be mathematicians. 2. The set of individuals who identify themselves as members of the mathematics community. 3. The set of individuals who belong to one or more professional mathematics organizations. 4. The set of individuals who teach, study, research, do, learn, or are interested in, mathematics. 5. The set of individuals whose job entails knowing and/or doing a significant amount of mathematics.

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Fig. 1 Logos of US professional mathematics organizations in CBMS

For the purposes of this chapter, the definition of the “mathematics community” to be used will be Definition 3, i.e., “The set of individuals who belong to one or more professional mathematics organizations.” There is an umbrella organization for several of the most prominent mathematics member organizations in the USA, and it is known as the Conference Board of the Mathematical Sciences (CBMS). The logos for the eighteen mathematics membership organizations in the USA that comprise and participate in CBMS are given in Fig. 1 and can be found at https:// www.cbmsweb.org/member-societies/. In Fig. 2 there is a depiction of the relative membership sizes of some of the professional mathematical organizations whose logos are depicted in Fig. 1. One can conclude from this 2017 data that the overall size of the US mathematical community as defined in this chapter is relatively small (less than 150,000 members). The names of the US mathematics membership organizations depicted in Fig. 1 and listed in Fig. 2 are given below (with links to their websites provided):

On the Diversity and Demographics of the US Mathematics Community

Fig. 2 Membership sizes of some US professional mathematics organizations

ASSM BBA ASL NAM TODOS AMTE AMATYC NCSM IMS AWM INFORMS MAA SIAM ASA SOA AMS NCTM

Association of State Supervisors of Mathematics Benjamin Banneker Association Association for Symbolic Logic National Association of Mathematicians TODOS: Mathematics for ALL Association of Mathematics Teacher Educators American Mathematics Association of 2-Year Colleges Leadership in Mathematics Education (formerly National Council of Supervisors of Mathematics) Institute of Mathematical Statistics Association for Women in Mathematics Institute for Operations Research and the Management Sciences Mathematical Association of America Society for Industrial and Applied Mathematics American Statistical Association Society of Actuaries American Mathematical Society National Council of Teachers of Mathematics

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These organizations can be grouped into different categories associated with their differing missions and segments of the mathematics community on which they focus. There are those that are primarily concerned with the teaching of mathematics (ASSM, BBA, NCSM, NCTM, AMTE), those that represent particular underrepresented demographic constituencies (NAM, AWM, TODOS), those that represent specific disciplines in the mathematical sciences (ASL, SOA, IMS, ASA, INFORMS, SIAM), and a few general purpose organizations (MAA, AMS).

3 Who “Does the Math”? In this section data on the demographics of mathematics degree holders in the USA is provided in order to underscore and document the underrepresentation by race, ethnicity, and gender of certain groups in the mathematics community. The data discussed in this section is publicly available from various sources. Information about undergraduate degrees primarily comes from two sources. First is the National Center for Educational Statistics (NCES) in the US Department of Education’s Institute for Education Sciences. The annual Digest of Education Statistics is a huge compendium of data about the educational enterprise in the USA at various levels [13–17]. The second source of data on undergraduate mathematics education is the quinqennial survey conducted by CBMS [18]. This survey has been conducted every 5 years since 1970; the results of the most recent survey conducted in 2015 were published in 2018 [19] and contain a plethora of information about undergraduate mathematics education (and the people who participate in it) in the USA. Information about doctoral degrees in mathematics in this section primarily comes from the Doctoral Recipients section [20] of the Annual Survey of the Mathematical Sciences [21] published by the American Mathematical Society. Figure 3 shows the percentage of undergraduate degrees in mathematics by the race of the recipients (White, Latinx, and African-American) from 2013 to 2018. The trend is clear: the percentage of white undergraduate mathematics degree recipients is decreasing more rapidly than the percentage of degrees to underrepresented students is increasing. Somewhat surprisingly, the percentage of Black mathematics graduates is decreasing very slowly, according to the given data, while the Latinx percentage is steadily rising. Figure 4 shows the percentage of undergraduate degrees in mathematics by the recipient’s gender from 2013 to 2018. The underrepresentation of female recipients of undergraduate mathematics degrees has remained almost constant during this time period, with barely a one point change in the percentage of mathematics undergraduate degrees earned by women in the interval for which data was obtained. Figure 5 shows the percentage of doctoral degrees in mathematics which have been earned by women from 2004 to 2017. The first thing to note about the data presented here is that the percentage of women Ph.D. recipients have remained in a narrow range between 25% and 33% for nearly 15 years. However, this means that women have been extremely underrepresented in the fraction of mathematics Ph.D.

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Fig. 3 Percentage of Bachelors degrees in mathematics by recipient’s race, 2013–2018

Fig. 4 Percentage of Bachelors degrees in mathematics by recipient’s gender, 2013–2018

recipients for decades. Clearly women are approximately 50% of the population so one could expect that approximately half of mathematics Ph.D. recipients would be female [22]. However, in order to obtain a Ph.D. one has to have received an undergraduate degree first. Comparing Figs. 4 and 5 makes visible that there is a significant drop from the percentage of undergraduate mathematics degree recipients who are female, which has been about 42.5% on average, and this is significantly higher than the approximately 27.5% of doctoral mathematics degree recipients who are female over the same period of time. This is a clear example of what is sometimes called the “leaky pipeline” in STEM, although some have noted that the effects appear to disproportionately harm women and racial minorities [23]. Since females make up a majority of the general population, clearly there is underrepresentation on the basis of gender at both the bachelors and doctoral levels.

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Fig. 5 Percentage of female Ph.D. recipients in Mathematics 2004–2017 Table 1 Race of US Citizen Female Mathematics PhD Recipients 2004–2017

Years 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

Black 1 9 5 5 11 16 9 9 10 6 9 10 11 11

Latina 5 6 11 4 5 12 8 9 11 6 7 9 11 6

White 118 101 102 132 161 154 168 155 163 170 179 195 201 189

Total 153 152 151 193 218 235 245 230 224 242 254 244 251 267

Tables 1 and 2 provide information about the number of Mathematics Ph.D. recipients by race and gender in the USA between 2004 and 2017. The source of this data is the American Mathematical Society’s Annual Survey of New Doctorates [21]. Please note that these numbers are not percentages but the actual raw totals of the number of individual US Citizens who earned Ph.D.’s in Mathematics for the relevant time period. These numbers are a stark testament to the degree of underrepresentation by African-American (Black) and Hispanic (Latinx) mathematicians in the USA. The numbers given in Tables 1 and 2 also demonstrate that the “leaky pipeline” regarding gender (visualized in Fig. 5 where the fraction of female mathematics doctoral degree recipients is much lower than the fraction of female mathematics

On the Diversity and Demographics of the US Mathematics Community Table 2 Race of US Citizen Male Mathematics PhD Recipients 2012–2017

Years 2012 2013 2014 2015 2016 2017

Black 16 16 15 10 18 19

Latino/a 22 22 24 17 34 27

9 White 492 522 564 545 551 527

Total 628 670 694 636 684 686

Fig. 6 Percentage of doctoral degrees in mathematics by recipient’s race or ethnicity, 2012–2017

bachelor degree recipients) becomes an almost completed “clogged” pipe when considering race and ethnicity. The percentages of Black and Latin mathematics Ph.D. recipients shown in Fig. 6 are significantly lower than the percentages of Black and Latinx people in the general US population (which are roughly 12% and 17%, respectively, [22]) as well as appreciably lower than the percentage of Bachelors degrees by recipient’s race depicted in Fig. 3. By either measure one can consider the underrepresentation of Black and Latinx mathematicians in the US mathematics community to be persistent and profound. The underrepresentation can be described as persistent because the numbers have not substantially changed in the last 5 years for which data is provided here. The phenomenon can also be described as profound because it has significant, wide-reaching effects on how mathematics and members of the mathematics community are perceived by others, how members of the mathematics community perceive themselves and how both of these perceptions can influence the teaching and learning of mathematics, by both people who are underrepresented and those who are overrepresented in the mathematics community [24–26].

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4 Conclusion In this section I will conclude this chapter with some comments on how the US mathematics community has responded to the existence of profound and persistent underrepresentation by race, ethnicity, and gender that has been demonstrated in earlier sections. In December 2019 a widespread debate over what kind of remedies and responses are appropriate to ameliorate underrepresentation by race, ethnicity, and gender in the mathematics community was sparked by a provocative editorial published in the most widely read mathematical periodical in the world, the Notices of the American Mathematics Society [27]. This article was published by Abigail Thompson, Professor (and Chair) of the University of California Davis mathematics department. In the piece, Prof. Thompson decried that her employer, the University of California, had decreed that all academic departments must use specific procedures to identify candidates interested in diversity, equity, and inclusion. Prof. Thompson compared this requirement to the odious “loyalty oath” which had been implemented at the height of America’s anticommunist fervor in the 1950s. To many members of the US mathematics community it appeared as if Prof. Thompson was expressing opposition to the idea of taking action to increase diversity and broaden participation by underrepresented minorities in mathematics, despite the fact that she began the document with a clear statement of support of such efforts (“We should continue to do all we can to reduce barriers to participation. . . ”) [27]. An analysis [28] of the several hundred letters that were written to and published in the next issue of the Notices of the AMS was able to discern and quantify differences in those who opposed and supported Prof. Thompson. Broadly speaking, those who opposed Thompson can be interpreted as demonstrating support for underrepresented individuals in mathematics, while those who supported Thompson can be interpreted as either supporting academic freedom to publish unpopular, controversial views or supporting the position that some actions have gone “too far” in supporting diversity and inclusion in mathematics. The analysis of the response to Thompson’s editorial demonstrates that there is a schism in the mathematics community around the issue of its diversity and demographics and what to do about it. Generally, the analysis [28] found that those who opposed Thompson were identified as less senior, more likely to be a member of an underrepresented group in mathematics and affiliated with “less prestigious” academic institutions. On the other hand, those who supported Thompson were overwhelmingly male with more established and longer careers at “more prestigious” institutions. Interestingly, the groups were almost evenly matched in size and number. To me the primary conclusion to be drawn from the entire controversy around l’affaire Thompson is that it is convincing evidence that the premise of this chapter is correct and that the issues discussed within are increasingly salient to many members of the mathematics community. Clearly, it is important “who does the math” or else hundreds of members of the US mathematics community (i.e., readers of the Notices of the AMS) would not have been inspired to publicly declare their

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allegiance to one side or the other on this topic. I hope that this energy can be channeled into action to address the ongoing underrepresentation by race, ethnicity, and gender in the mathematics community. Acknowledgments The author would like to acknowledge financial support from Occidental College for travel to attend the Ninth International Congress on Industrial and Applied Mathematics (July 15–19, 2019 in Valencia, Spain). I owe a huge debt of gratitude to Professor Nicole Josephs of Vanderbilt University (https:// peabody.vanderbilt.edu/bio/nicole-joseph) for first alerting me to the existence and salience of the data given in Tables 1 and 2.

References 1. Jacobs, H.R.: Mathematics, A Human Endeavor, 3rd edn. W.H. Freeman, New York (1994) 2. Abfouadel, E., Braddy, L., Carpenter, J., Douglas, L., Gillman, R.: The importance of mathematical sciences at colleges and universities in the twenty-first century. Math. Assoc. Am. (2018). Cited 23 November 2019. https://www.maa.org/node/1566047 3. Prepare and Inspire: K-12 Science, Technology, Engineering, and Math (STEM) Education for America’s Future. https://obamawhitehouse.archives.gov/sites/default/files/microsites/ ostp/pcast-stem-ed-final.pdf. Cited 23 November 2019 4. Engage to Excel: Producing One Million Additional College Graduates with Degrees in Science, Technology, Engineering, and Mathematics. https://obamawhitehouse.archives.gov/ sites/default/files/microsites/ostp/pcast-engage-to-excel-final_2-25-12.pdf. Cited 23 November 2019 5. Successful K-12 STEM Education: Identifying Effective Approaches in Science, Technology, Engineering, and Mathematics. http://www.stemreports.com/wp-content/uploads/2011/ 06/NRC_STEM_2.pdf. Cited 23 November 2019 6. National Academy of Sciences (NAS): Rising above the Gathering Storm: Energizing and Employing America for a Brighter Economic Future. Washington, DC, National Academy Press (2012) 7. Bybee, R.W.: The Case for STEM Education: Challenges and Opportunities. Arlington, VA, NSTA Press (2013) 8. Kennedy, T.J., Odell, M.R.L.: Engaging Students in STEM Education. Sci. Educ. Int. 25(3), 246–258 (2014) 9. About the National Science Foundation. https://www.nsf.gov/about/. Cited 21 January 2020 10. About Education and Human Resources (EHR). https://www.nsf.gov/ehr/about.jsp. Cited 21 January 2020 11. NSF Budget Requests to Congress and Annual Appropriations. https://www.nsf.gov/about/ budget/. Cited 21 January 2020 12. Su, F.: Mathematics for Human Flourishing. London, Yale University Press (2020) 13. Snyder, T.D., de Brey, C., Dillow, S.A.: Digest of Education Statistics 2014 (NCES 2016-006). National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education, Washington (2016) 14. Snyder, T.D., de Brey, C., Dillow, S.A.: Digest of Education Statistics 2015 (NCES 2016-014). National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education, Washington (2017) 15. Snyder, T.D., de Brey, C., Dillow, S.A.: Digest of Education Statistics 2016 (NCES 2017-094). National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education, Washington (2018)

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16. Snyder, T.D., de Brey, C., Dillow, S.A.: Digest of Education Statistics 2017 (NCES 2018-070). National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education, Washington (2019) 17. Snyder, T.D., de Brey, C., Dillow, S.A.: Digest of Education Statistics 2018 (NCES 2020-009). National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education, Washington (2020) 18. CBMS Survey. Conference Board of the Mathematical Sciences. http://www.ams.org/ profession/data/cbms-survey/cbms-survey. Cited 25 April 2020 19. Blair, R., Kirkman, E.E., Maxwell, J.W.: Statistical abstract of undergraduate programs in the mathematical sciences in the United States: Fall 2015 CBMS survey. In: Conference Board of the Mathematical Sciences (2018) 20. Doctoral Recipients. Mathematical and Statistical Sciences Annual Survey. American Mathematical Society, New York (2019). https://www.ams.org/profession/data/annual-survey/phdsawarded. Cited 23 November 2019 21. Mathematical and Statistical Sciences Annual Survey. American Mathematical Society, New York (2019). https://www.ams.org/profession/data/annual-survey/annual-survey. Cited 23 November 2019 22. U.S. Census Bureau: American FactFinder 2013–2017 American Community Survey 5Year Estimates. https://factfinder.census.gov/faces/tableservices/jsf/pages/productview.xhtml? src=CF. Cited 23 November 2019 23. Blickenstaff, J.C.: Women and science careers: leaky pipeline or gender filter. Gender Educ. 17(4), 3690–386 (2005) 24. Williams, M.J., George-Jones, J., Hebl, M.: The Face of STEM: Racial Phenotypic Stereotypicality Predicts STEM Persistence by— and Ability Attributions about—Students of Color. J. Pers. Soc. Psychol. (2018). https://doi.org//10.1037/pspi0000153 25. Bressoud, D.: Persistence of Black and Latino/a Students in STEM 26. Gutiérrez, R.: Political Conocimiento for Teaching Mathematics: Why Teachers Need it and how to Develop it. In: Kastberg, S.E., Tyminski, A.M., Lischka, A.E., Sanchez, W.B. (eds.) Building Support for Scholarly Practices in Mathematics Methods. The Association of Mathematics Teacher Educators (AMTE) Professional Book Series, pp. 11–38. Information Age Publishing Inc, Charlotte (2017) 27. Thompson, A.: A word from. . . . Am. Math. Soc. 66(11), 1778–1779 (2019). https://www.ams. org/journals/notices/201911/rnoti-p1778.pdf 28. Topaz, C., Cart, J., Eaton, C.D., Shrout, A.H., Higdon, J.A., Ince, K., . . . , Smith, C.M.: Comparing demographics of signatories to public letters on diversity in the mathematical sciences. PloS one 15(4), e0232075 (2020). https://doi.org/10.31235/osf.io/fa4zb

Teaching Calculus as a Tool in the Twenty-First Century Emma Smith Zbarsky, Gary Simundza, and Mel Henriksen

Abstract The calculus curriculum has seen remarkably little change over time even as the computational power available to students has increased. The Applied Mathematics department at the Wentworth Institute of Technology has been working to actively address this issue for the past 4 years. We will discuss the curriculum currently used for a year-long calculus sequence for students who will apply their knowledge in engineering and computer science, as well as the compromises and areas of opportunity we have found along the way. The major components of our approach include a series of inquiry-based in-class group exercises and six group projects. The in-class activities provide an initial review of prerequisite material and introduce students to major concepts in both differential and integral calculus. This in-class work is reinforced and extended in the group projects. Each project culminates in a formal paper and/or an oral presentation. A streamlined textbook, developed especially for this course by the authors and their colleagues, provides a resource for the students and includes relevant homework assignments and workedout examples.

1 Background and Significance The content of introductory calculus courses [1, 2] has been largely unchanged over the last century despite significant changes in the use of calculus in applied fields. The course begins with limits, moves into analytic derivatives of common functions, as well as the product and chain rules for derivatives, and offers some geometrical supporting reasoning for this. There will be some discussion of Taylor series and at some point some discussion of integrals and a short section on finding extrema. The majority of the book, and frequently of the course, will be spent on long and involved calculations of particular derivative and integral problems such as computing the

E. S. Zbarsky () · G. Simundza · M. Henriksen Department of Applied Mathematics, Wentworth Institute of Technology, Boston, MA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 R. Buckmire, J. M. Libertini (eds.), Improving Applied Mathematics Education, SEMA SIMAI Springer Series 7, https://doi.org/10.1007/978-3-030-61717-2_2

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tangents to a cycloid or the Witch of Agnesi [3]. In the late nineteenth and early twentieth centuries, this included a lot of analytic geometry, while a modern calculus sequence will likely spend much time on methods of integration with brief discussions of applications such as calculating work or finding the center of mass, and some time spent with solving simple differential equations, c.f. [4, 5]. The tone of introductory calculus courses is frequently an odd mix of formal mathematics and slipping difficulties under the rug. George Berkeley complained about this in 1734 with his “ghosts of departed quantities” in The Analyst, [6]. Leaving aside courses aimed at presenting calculus for “less mathematical” disciplines, modern calculus textbooks can vary from the “calculus reform” that Hughes-Hallett [7] used at the University of Michigan to popular Stewart [8] or rarefied Apostol [9]. At the same time, many students attempt to circumvent the work of learning by applying technology, whether that is to use a solution book paired to the latest copy of their textbook [10] or to simply request Step-by-Step Solutions from Wolfram Alpha [11]. This challenge has long been recognized, most commonly under the label of “reform calculus” in the USA, but it is being grappled with in Europe as well [12]. Our approach is related to reform calculus insofar as we are eliminating some traditional topics from the syllabus and incorporating significant amounts of group “laboratory” work into the course. We differ in that we are including a much more significant discussion of numerical methods and the applications of the ideas and methods of calculus to data, as well as in the fact that we are very upfront with our students about the fact that this course is intended to teach practical calculus. While we make every effort to avoid mathematical falsehoods, this results in frequent acknowledgement of the need to delve into analysis for complete understanding of potential pathological examples or other counterexamples beyond the scope of the functions we are discussing. The challenge running through the last century of calculus instruction is the question of what students need to know in a field that is both vast in application and with 300 years of history in more-or-less the current form, and yet where the core ideas are quite simple: rates of change and accumulation. We have found that our students are frequently having trouble with correctly performing algebraic manipulations. There is a lot of intuitive calculus that can be considered with no algebra at all [13]. Consider, for example, the relationship between position, acceleration, and velocity presented and interpreted graphically. This must be balanced against the fact that about 50% of our students have taken a calculus class before they reach us, mostly in high school. Our university awards credit for scoring at least 3 (out of 5 possible) on the national “Advanced Placement” series of exams in calculus, so most of these students are not comfortable with calculus despite their exposure. A score of 3 is intended to be equivalent to earning a C (notable bajo) in a college-level course. The (applied) math department at the Wentworth Institute of Technology decided to take up this challenge 4 years ago. We spent a year with blue-sky thinking about what mathematics we want our graduates to understand as well as some realistic analysis of what mathematics our matriculating students are comfortable

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with. We asked ourselves how the mathematics in courses had changed before, especially when driven by technological changes such as the use or elimination of the slide rule. Building on this work, three of us sat down and designed a new year-long calculus sequence for entering engineering, computer science, and applied mathematics majors that is now being taught to about 200 students per year. The 2019/2020 academic year will be our third class of students. We aim for four areas of understanding: conceptual fluency, basic analytic computational fluency, basic numerical computational fluency, and the willingness and ability to translate real situations into tractable mathematical formulations and then solve them. As instructors, we aim to view technological tools as an everyday part of our learning environment and to design the course to use them as an aid to understanding rather than a barrier. The resulting year-long course contains six interdisciplinary projects that we developed to build flexible real-world thinking about mathematics. To ensure computational fluency, we require reaching 80% on a 20-question test of basic competencies with integration and differentiation. To address conceptual fluency, we work through conceptual problems in class and ask conceptual questions on midterms and final exams.

2 A Note On Rigor In our applied calculus curriculum, we are consciously and thoughtfully eschewing analysis in favor of intuitive understanding of common problems in calculus. As Kalid Azad writes, “Would it be so bad if everyone understood calculus to the ‘non-rigorous’ level that Newton did?” [14]. To this end, we limit our library of functions that students are expected to be fully fluent with to just five: sin(x), cos(x), ex , ln(x), and x n for any real n, as well as any functions created through linear combinations, multiplication, division, and/or composition of these functions. We discuss limits graphically and numerically using computer support, with a minimum of algebra and an emphasis on common continuous functions, removable discontinuities, and asymptotes. The historically difficult sticking points have been smoothed out dramatically with computational tools such as: • • • •

Desmos (https://www.desmos.com/), WolframAlpha (https://www.wolframalpha.com/), MATLAB (https://www.mathworks.com/products/matlab.html), Spreadsheet programs such as Excel or Google Sheets.

While most programs in calculus are fairly similar, we did take inspiration from aspects of the innovative programs at the US Military Academy (early differential equations), the Arizona State University (Project DIRACC), and the University of Michigan (team work and competency exams) [15–17].

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3 Course Overview We divide the material into two courses, each delivered over 15 weeks of time including examination periods. The course officially meets for 260 min per week with two 75-min meetings and one 110-min meeting. Each section is capped at approximately 25 students. Each student is provided with a university-issued laptop and licenses for all non-free software, such as MATLAB, that is used in this course. In the first semester, we begin by discussing and practicing data collection and analysis including linear regression. We continue on to a review of functions including polynomials, exponentials, logarithms, sines, cosines, tangents as well as rules for creating more complex functions via linear combinations, multiplication and division, and composition of functions. At this point, we focus on slopes and rates of change leading into the limit definition of the derivative after several laboratory exercises investigating numerical derivatives. We also quickly introduce higher-order derivatives and differential equations. This leads into the first major project in the course, which will be discussed in detail below, on differential equations. Following these definitions, while students are working on their differential equations project, we move into discussing formulas for calculating derivatives. After completing the differential equations project, we use Euler’s method to segue into accumulation and integration. We begin by revisiting our original numerical derivative velocity calculations and checking that we can calculate the same incremental differences in our displacement using either positions and times or velocities and times. This brings us to approximating areas under curves, and thus into integrals–our second project. Antiderivatives are introduced with a focus on their application to solving differential equations. We finish the first semester with a discussion of related rates, and an accompanying project. In the second semester, we begin by reviewing the relationships between curves, accumulation, and rates of change by discussing optimization, including the fourth project. Then we move into substitution as a method of integration. This is covered quite intensively since for all standard functions that we do not include in our course library, the students are expected to integrate and differentiate them using tables. We selected a standard table of integrals and derivatives that is used on the Fundamentals of Engineering exam, the standard professional licensing exam for all engineers in the USA. Outside of school, such a standardized exam is likely to be the only time our students will be asked to solve calculus problems without reference to their chosen computational devices. At this point, we briefly cover integration by parts, integration using trigonometric identities, and integration of rational functions using partial fractions to explain how the entries in the table of integrals were created. At this point, the students take their “Absolutely Basic Competencies” exam. The course moves on to discuss geometric applications of integrals and the fifth project. Finally, the course concludes with a discussion of approximating functions. We move from the linear approximations we used for Euler’s method in the first term to quadratic or cubic approximations and then extend to Taylor series and the final course project.

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4 Opening Activities On the first day of the course, the students measure the length of their forearms and their heights and create a class scatterplot by hand. After estimating a linear fit by hand, they are provided with instructions to compute a least squares regression line using Excel, MATLAB, and R. They use their class data to argue whether they believe the forearm bones located on Nikumaroro Island in 1940 could belong to Amelia Earhart. This allows us to briefly discuss error, precision, and sampling bias—for instance, is it reasonable to use a male-skewed sample of about 20 18year-old twenty-first century students to reason about the physical structure of an early twentieth-century adult female? We also use this to reinforce the idea that all of the mathematics that we are doing can be done by hand—the first textbook reading works through the formulas for computing a least-squares regression line— but that it is frequently more efficient to use a computer for repetitive calculations. The next activity is to collect data on the buckling point of spaghetti at varying lengths and use the data collected as a reason to review the basic properties of the standard library of trigonometric functions, exponentials, logarithms, polynomials, and rational functions including their horizontal and vertical asymptotes, general shape, and growth tendencies. Students who enter our class with a traditional calculus background frequently start wondering aloud where all the formulas are that they are expecting around this time. We encourage them to look for the concepts they were expecting in the next laboratory experiment. As we move into derivatives, we have the students work in teams of three to collect data from two videos: the first is a basketball in parabolic motion and the second is a damped oscillation in a cantilevered spaghetti noodle (see Fig. 1). In

Fig. 1 Data collection from a video of a cantilevered spaghetti noodle at 240 frames per second

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Fig. 2 Fitting data and first-order difference quotient data for the noodle shown in Fig. 1

each case, they use LoggerPro to collect a position value for each frame in the video. After copying their raw data into a spreadsheet program, they clean up their data, for instance by eliminating data collected before motion began in each video or by resetting their origin position in time and/or space. Using the difference quotient, they calculate numerical derivatives. They graph their data and fit an appropriate curve to it in Desmos (see Fig. 2). Students who have studied physics tend to notice that the slope of the line for the parabolic motion is around −9.8. For the cantilevered noodle, students generally recognize that if they fit a sine curve to their position data, then a cosine curve with the same frequency fits their velocity data. They generally do not recognize that the amplitude of the velocity curve has been multiplied by the frequency and that must be pointed out by the instructor. While they subsequently fit a product of a decaying exponential and a sine curve to better approximate their data, they do not recognize the product rule as governing their velocity data until after we have covered it in theory and then return to the dataset to check it in application.

5 Differential Equations Project For the first project, students are placed in new groups. Initially they all investigate two population models during classtime, one of rabbits in the Boston area based on a Boston Globe newspaper article and the second on fitting the human population of the USA. This allows them to investigate the basic properties of dynamical systems such as stable and unstable equilibria. They are also introduced to ode45, which is an implementation of a Runge–Kutta explicit numerical method in MATLAB, which allows them to see exponential growth, exponential decay, and the logistic growth

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model appearing for different differential equations and choices of parameters and initial conditions. For the group project, primarily completed outside of class, the students are provided with a selection of possible differential equation models to explore including SIR models, spring–mass systems with friction, drug absorption and elimination models, radioactive decay cascades, LRC circuit models, and Lotka–Volterra population dynamics models. Each group explores their model and formulates their differential equation(s), solves the equation(s) in MATLAB for a range of different parameter values, and presents their findings in a paper and as a formal presentation to the entire class. Each group is provided with a rubric to guide their paper write-up and formal presentation of their results. Each presentation is marked on organization, presentation, and answers to questions, and there is an individual score component for each group member. Individuals are scored down both for talking too much and for talking too little or for indications that they do not understand the work they are presenting. The write-up for this project is scored on background, solution, results and analysis, conclusions and reflections, bibliography, and writing.

6 Numerical Integration Project For the second project, students are placed in new groups. During the in-class time for this project, they all numerically compute integrals of a fifth degree polynomial using a midpoint approximation, a trapezoidal approximation, and with a Simpson’s approximation that is a weighted average of the first two. They repeat these estimates with 4, 8, and 16 subintervals and compute the error in each case. Outside of class, they locate a pond of their choosing, usually through the US Geological Survey website or through Google Maps. They are instructed to use numerical methods to estimate the surface area of their chosen pond and discuss the accuracy of their result. This gives them a chance to practice thinking about how these methods interact outside of a standard instructor-provided sterile function. It is common that misunderstandings about what Simpson’s rule is actually doing appear at this point. Students also end up grappling with precision choices. Only a few groups realize that the tools we taught them earlier, such as LoggerPro, can be very useful in computing the measurements that they want of images acquired online. Every group takes advantage of spreadsheets for their data organization and computation, however. The presentations are quite interesting because the students quickly realize that certain aspects of their pond problems, such as how to deal with boundaries and the fact that Simpson’s method is claimed to give the least error, are common across everyone’s projects, but the analysis of potential or known error can be quite illustrative. This is also a good project to keep working on the students’ abilities to ask incisive questions of presenters since each project is unique but with clearly recognizable, overlapping mathematical methods applied in each case. The prose write-up for this project is generally only a couple of paragraphs explaining why the group made the choices that they did and how those

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choices impacted the error in their results. The rest of the write-up grade is based on the correctness and completeness of the calculations made in the spreadsheet implementing the midpoint method, trapezoidal method, Simpson’s method, and applying some justified numerical approximation method to their chosen pond.

7 Related Rates Project For the third project, students are frequently allowed to select their own groups with moderate supervision by the instructor. At this point, they have worked with many of the students in the course, and they are starting to learn which combinations of work habits, work ethic, course schedules, and evening/weekend availability are the optimal choices for them. We do have a mix of dorm-resident (on-campus) and commuter (off-campus) students. They frequently sort themselves first along those lines when given the option. This project begins with a common analysis during class of the rate of change in the height of liquid flowing into flasks of several different shapes: cylindrical and round-bottom, typically. Then each group can select a related rates project to investigate on their own, largely outside class, from among a collection including: modeling the speed of a piston, modeling an oscillating sprinkler, modeling the creation of a lake following the building of a dam, and modeling the filling of an erlenmeyer flask. In this case, each project option comes with one page of background information and questions to be answered. Again, each group works through their model and writes a paper answering all the posed questions in prose style. They also present their work to the class. In this project, we encourage each group to highlight areas where they had the most difficulty. Some of these projects have intricate arithmetic, like the erlenmeyer flask. Some require particular care with trigonometric identities, such as the analysis of the piston or with keeping careful track of units, such as the piston model. The write-up for this project is graded on background, static equation(s), dynamic equation(s), conclusions and reflections, answers to project choice-specific questions, and writing.

8 Optimization Project For the fourth project, students are just learning their way with new classmates and a new instructor. However, the basic procedures for this course have been set so this project has a brief set of common instructions: Choose one of the following problems and conduct a thorough analysis of it. Identify an objective function, and use methods of differential calculus as discussed in the class activity and Chapter 9 to optimize it. Your report should include the following:

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• a brief introduction to the problem, • a complete description of your analysis, including how you found your objective function and how you used derivatives to optimize it, • an explanation of how you know your result is a maximum (or minimum), including a second derivative test, • any specific additional results requested in the problem statement, • a summary/conclusion and a list of any references consulted, • a statement about meeting times and group interaction.

The options for this project include optimizing a blood vessel branching model, a wind turbine model, a resonant frequency model, and a machining operation model. The provided rubric notes that students will be graded on their background and objective function, optimization calculation, results, graphs, and analysis, conclusions and reflections, bibliography, and writing.

9 Geometric Applications of Integrals Project For the fifth project, students have learned all the analytical rules for integration and differentiation that we cover and have already completed, in most cases, the Absolutely Basic Competencies exam demonstrating computational fluency, as described in Sect. 11. In this project, we ask them to engage with the extension of these core calculus ideas in three dimensions. They are given several possible design projects including analyzing classic dome construction, investigating a dam, or exploring the forms in a proposed hotel to be built into a local quarry. The last project is particularly exciting as the students are given a copy of an old Boston Globe newspaper article (see Fig. 3) regarding this proposal and asked to estimate several of the volumes and surface areas referenced in the construction plans. They have very little information other than a couple of dimensions given in the article and an artist’s sketch of the proposed building. It is fascinating to see how each group makes slightly different assumptions but how they can all still check each Fig. 3 A hotel inspired by a newspaper article published in the Boston Globe on June 2, 1990

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other because any reasonable assumption will result in solutions of a similar order of magnitude. When a group makes a mistake in their calculations and the others start to present, they quickly realize that they must have gone wrong somewhere if their results differ by two orders of magnitude. We generally offer about one class period total, but spread out over a full week, to working on this project. That allows groups to run their initial assumptions by the instructor before they have gone too far with oversimplified or overcomplicated geometrical calculations. The write-up for this project is graded on background and introduction, assumptions and solution development, calculus setup and computation, conclusions/summary and reflections, overall structure, and writing.

10 Taylor Series Project For the final project in this sequence, the students are invited to explore the capabilities of Taylor series. Our students tend to have 2–3 term projects due in other courses at the same time, so we generally offer two class meetings devoted to working on this project so the mathematics can be completed with instructor oversight. One potential project leads them through using Taylor series to estimate the necessary interest rate, given a fixed deposit and a known goal over a given timespan. Another project leads the students through the ideas of finite difference method approximations and encourages them to recognize that the first-order numerical approximation to a first derivative that they have been using throughout the year is not the only numerical approximation possible and in fact they can easily create other approximations with varying orders of error by applying Taylor’s theorem. Both of these projects offer more in the way of organized direction than the earlier projects in the sequence. This results in less interesting technical presentations at the end, which requires the students to focus on their communication skills, thoughtfulness in slide design, and thoroughness of explanation. The write-up is scored on the correctness of solutions to the problems presented (70%) and the quality of the writing (30%).

11 Absolutely Basic Competencies Exam The Absolutely Basic Competencies (ABC) exam is a 20-question exam covering graphical understanding of derivatives and integrals, position/velocity relationships, taking derivatives using the product rule, the chain rules, linearity rules, and the derivatives with respect to x of x n , ex , ln(x), cos(x), and sin(x), where n is explicitly chosen to be positive, negative, and fractional/irrational in different problems. There is one problem involving computing a second or a third derivative. They also must integrate using linearity, substitution, and the integrals of x n , n = −1, ex ,

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1 , cos(x), and sin(x). They must solve a simple initial value problem such as x dy = 2(x − 2)−3 with y(0) = 1. They must also apply the fundamental theorem dx of calculus to compute at least two definite integrals. We have traditionally created these exams in their entirety and given them on paper. This allows us to carefully balance the appearance of each type of function across the exam without writing complex algorithms and without requiring each type of problem to always involve a particular type of function. Because of overlapping timing, we are generally able to write only 6–12 versions of this exam each year even if we are administering it to 600 students repeatedly. That said, as the numbers of students go up, creating an electronic database, even if it will only generate new paper versions of such an exam is desirable. Our current methods are a simple version of this with one .tex file containing 10–15 different versions of each type of problem we want on the exam. The ABC exam is graded with minimal partial credit. If the calculus is wrong, there is no credit given. If the calculus is correct but after the calculus was computed there was an arithmetic or copy error introduced during simplification, partial credit of 4 out of 5 (or 3 out of 5 for egregious arithmetic mistakes) is awarded. A score of 80/100 is required to pass. If a passing score is not earned on the first attempt, the score recorded in the gradebook is capped at 80% when the exam is eventually passed.

12 Evidence of Effectiveness Over the past 2 years, all freshman computer science students took our new calculus sequence, “Integrated Engineering Calculus I and II.” A formal study comparing the “traditional” and “integrated” sequences is planned for the next academic year, but initial data have been quite positive. On Jerome Epstein’s Calculus Concept Inventory, one professor’s students have a much steadier improvement rate of 15.9% in both 2017 and 2018, while the same professor’s traditional students historically scattered from 7.2% in 2014 to 20.0% in 2015 and 14.0% in 2016. In the first term, computational abilities are slightly lower with an average final exam score of about 57% in the integrated version and 68% in the traditional version, as measured by common questions in the final examination. In the second semester of the integrated sequence, there is a (repeatable) test of basic computational competencies that mirrors a pair of (repeatable) tests of basic computational competencies in the traditional version that has similar pass rates with more than 90% of the students scoring 80% or better. Over the past 2 years, the D/F/W rate–the rate of students not completing the course at a satisfactory level–has been almost identical at 20.7% in the first term of the integrated sequence compared to 20.4% for the traditional version, but we expect that this is partially accounted for by the differing instructors with a large percentage of generously grading adjuncts in the traditional sequence. Anecdotally, the students in the integrated sequence are far more comfortable with

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ambiguity, calculus on data and nontraditional variable names. In fact, when asked what the big idea of calculus was on the first midterm examination, one student wrote: The big idea of calculus is [to] find the relationship between one idea and another. You can use something like position to measure the velocity at a certain point on a graph and vice versa. Calculus [. . . ] appears in problems with rates mainly, and find[ing] the relationship between the rates can lead to more data.

13 Future Directions We plan to update the scoring of the ABCs exam to incentivize continued improvement and emphasize the need for computational fluency by requiring a minimum of three retakes of the exam. The grade recorded would be the final score, if better than the previous scores, or the average of the previous and current scores. As we move engineers into this course, we will need to emphasize the distinction between discrete difference equations and the continuous models that we create for ease of computation more than we currently do for the computer science students. In addition, we will need to add a discussion of L’Hôpital’s rule for computing limits. We anticipate doing so toward the end of the course so that we can use the work with Taylor series to motivate the conclusions. We have also been actively engaging with engineering faculty members to help us identify additional projects based in real problems arising in their fields.

References 1. Thompson, S.P.: Calculus Made Easy. Project Gutenberg (2010). https://www.gutenberg.org/ ebooks/33283. Cited October 14, 2019 2. De Morgan, A.: Elementary Illustrations of the Differential and Integral Calculus. Project Gutenberg (2012). https://www.gutenberg.org/ebooks/39041. Cited October 14, 2019 3. Edwards, J.: An Elementary Treatise on the Differential Calculus With Applications and Numerous Examples. Macmillan, London (1896) 4. Department of Mathematics: Math 125 Materials Website. University of Washington, Washington (2017). https://sites.math.washington.edu/~m125/. Cited October 14, 2019 5. Dang, H.: Math 1310-001: Calculus I Fall 2017 Syllabus. University of Virginia, Virginia (2017). https://people.virginia.edu/~hqd4bz/Math%201310%202017F.pdf. Cited October 14, 2019 6. Berkeley, G. : The Analyst; or A Discourse Addressed to an Infidel Mathematician. Wikisource (2016). https://en.wikisource.org/wiki/The_Analyst:_a_Discourse_addressed_to_an_ Infidel_Mathematician. Cited October 14, 2019 7. Hughes-Hallett, D., et al.: Calculus: Single Variable. 7th edn. Wiley, New York (2019). ISBN: 978-1-119-53723-6 8. Stewart, J., et al.: Calculus: Early Transcendentals. Cengage, Boston (2016). ISBN: 978-1-28574155-0 9. Apostol, T.: Calculus, 2nd edn. Wiley, New York (1991). ISBN: 978-0-471-00005-1

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10. Slader: Homework Solved: Solutions to Stewart Calculus, 7th edn. Slader (2019). https://www. slader.com/textbook/9780538497817-stewart-calculus-7th-edition. Cited October 14, 2019 11. WolframAlpha: Step-by-Step Solutions. WolframAlpha LLC. https://www.wolframalpha.com/ pro/step-by-step-math-solver.html. Cited October 14, 2019 12. Törner, G., Potari, D., Zachariades, T.: Calculus in European classrooms: curriculum and teaching in different educational and cultural contexts. ZDM 46(4), 549–560 (2014). doi:10.1007/s11858-014-0612-0 13. Vangelova, L.: 5-Year-Olds Can Learn Calculus—The Atlantic (2014). https://www. theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124. Cited October 20, 2019 14. Azad, K.: A Gentle Introduction to Learning Calculus. Better Explained (2019). https:// betterexplained.com/articles/a-gentle-introduction-to-learning-calculus. Cited October 20, 2019 15. Dungan, I., Boucher, R.: A deviation from the long gray line an alternative calculus sequencing. MAA Focus 37(4), 16–17 (2017) 16. Thompson, P.W., Ashbrook, M.: Calculus: Newton, Leibniz, and Robinson meet technology. Project DIRACC (2019). https://patthompson.net/ThompsonCalc/. Cited November 3, 2019 17. Department of Mathematics: Math Introductory Program Student Guide. University of Michigan, Michigan (2019). https://www.math.lsa.umich.edu/courses/sg/. Cited November 3, 2019

Engaging Students in Applied Mathematics Education and Research for Global Problem Solving Wei Wang and Padmanabhan Seshaiyer

Abstract In this chapter, we present a novel way to engage students in applied mathematics education and research by combining context provided by the United Nations Sustainable Development Goals that helps to drive content demonstrated via a new mathematical modeling cycle for global problem solving. To illustrate the approach, we select a specific goal in this chapter, namely, health and well-being (SDG-Goal 3) and apply the eight different phases of the mathematical modeling cycle that consists of Observe, Theorize, Formulate, Describe, Analyse, Simulate, Validate and Predict to solve the global problem of controlling the spread of the Zika virus. Along with these phases, we also present the United Nations Sustainable Development Goals as a context to motivate the mathematical modeling cycle as a powerful mechanism to both engage students in a global problem-solving process and also excite them to pursue applied mathematics education and research.

1 Introduction When a community of educators and university students are polled on what they think will be the top response from the Google search engine as a phrase to complete the prompt “Education makes me”, the most common response from the community is the word “smart”. On the contrary, it is surprising to most when they find out the first response that one of the most powerful search engines such as Google returns is “Education makes me depressed”. To complicate matters, the first set of responses that the Google search engine returns for “math makes me” are cry, angry and stupid. Related to this observation, there have also been continuous debates among educators and scientists on the relevance of learning and motivation in education over the years [1]. Scientists who work on motivation research have continued to

W. Wang · P. Seshaiyer () Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 R. Buckmire, J. M. Libertini (eds.), Improving Applied Mathematics Education, SEMA SIMAI Springer Series 7, https://doi.org/10.1007/978-3-030-61717-2_3

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challenge this notion of relevance and have directed considerable empirical attention to create interventions to help students make meaningful real-world connections to what they learn in their classrooms. This topic of relevance has also been discussed by several researchers with reference to the teaching and application of mathematics [5, 12, 13, 28]. While some of these research have shown that targeted interventions fostered more positive value beliefs for mathematics among students, there is still a lot of work that needs to be done to engage student learning through applications of mathematics to the real world. The Common Core’s fourth Standard [9] for Mathematical Practice: Model with mathematics suggests that “mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society and the workplace.” A related framework has been proposed by several researchers across multiple levels including K-12, and undergraduate education is mathematical modeling [4, 29, 32, 33, 38]. National Council of Teachers of Mathematics Principles to Actions: Ensuring Mathematical Success for All has also provided examples of using the real world as a way for thinking about mathematical concepts [19]. The focus in these works has mostly been on the pedagogy of mathematical modeling in local contexts so that students can learn to see mathematics in their lives and cultural backgrounds as well as connect to the real world. Other researchers have focused their work around combining mathematical modeling with differential equations and technology to provide opportunities for undergraduate students to dive into research on the applications of mathematics to the real-world problems [20, 27, 28, 30]. In this paper, we propose that mathematical modeling can help to engage students in applied mathematics education and research for global problem solving.

2 Mathematical Modeling and Problem Solving Polya’s How to Solve It introduces a critical four-step approach as a framework for problem solving [22]. These steps included Understand the problem; devise a plan; Carry out the plan; and look back. These simple four steps enable students to plan, think, test and verify, respectively, as they solve a problem. As a parallel to this, there have also been several developments on how mathematical modeling may provide a way to help K-12 students understand problem solving. Specifically, mathematical modeling has been successfully introduced in K-12 as a powerful tool for developing students’ twenty-first century skills advancing their conceptual understanding of mathematics and developing their appreciation of mathematics as a tool for analysing critical issues in the world outside the mathematics classroom [31, 34]. It provides the opportunity for students to solve genuine problems and to construct significant mathematical ideas and processes instead of simply executing previously taught procedures and is important in helping students understand the real world [10]. It must also be pointed out that there have been active discussions and debates on how problem solving may be distinct from modeling in terms of task selection, process followed and implications for research [37].

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While there is a lot of focus on connections of mathematical modeling and problem solving for K-12 education, there is not sufficient literature in undergraduate applied mathematics education on how to engage students in making the connections between mathematical modeling and problem solving. This is often the case because most examples used in the textbooks are too ideal to represent the authenticity of being a real-world problem. So how do we give an authentic real-world problem-solving experience to undergraduates in applied mathematics education? Where do we find these authentic problems addressing global challenges? And, how do we solve them using mathematical modeling? Moreover, we must provide students the opportunity to select their own topic of research and then identify the mathematical models and tools they need to solve the problem. If this context for choosing the problem is global, that makes it all the more interesting for students. One such framework we employed with students was provided in 2015 by the United Nations when they set up the Sustainable Development Goals 2030 (SDG2030) agenda [35]. This has now provided a blueprint for shared prosperity in a sustainable world. This document that identified 17 Goals provides a comprehensive platform from selecting global challenges. These goals called for concrete efforts toward building an inclusive, sustainable and resilient future for people and planet. The vision was that through this SDG2030 agenda we can help eradicate poverty; promote sustainable, inclusive and equitable economic growth; create greater opportunities for all; reduce inequalities; raise basic standards of living; foster equitable social development and inclusion; and promote integrated and sustainable management of natural resources and ecosystems. Next, we will describe an example of an applied mathematics research project that aligned well with the third goal of SDG2030: Ensure healthy lives and promote well-being for all at all ages and how we employed a new mathematical modeling cycle described next to solve a global challenge that addressed this SDG goal. Note that there are several different frameworks that have been presented for describing mathematical modeling by various researchers over the years. For example, the GAIMME report [14] presented components of the modeling process to include identification of the problem; making assumptions and identifying variables, doing the math; analysing and assessing the solution; and iterating and implementing the model. In this paper, we present an alternate to this modeling cycle that expands these components to eight specific sub-modeling phases including: Observe, Theorize, Formulate, Describe, Analyse, Simulate, Validate and Predict. This is illustrated in Fig. 1. Note that one goes back to the observe phase as the mathematical modeling cycle is assumed to be an iterative approach. For illustrating the various phases of mathematical modeling shown in Fig. 1, next we will consider the application of the spread of infectious diseases as a context that students can easily relate to. This global challenge aligns well with SDG2030 goal 3 and admits techniques from applied mathematics for addressing the dynamics of the spread of infectious diseases. Moreover, the exposition presented in the various mathematical modeling phases builds on a variety of contributions in the area of infectious diseases over the years and references therein [2, 6, 16, 26]. The

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Fig. 1 A framework for mathematical modeling

purpose of the discussion in this chapter is to highlight the importance of the eight phases when doing mathematical modeling.

2.1 The Observe Phase The process of modeling involves multiple phases and starts with the observe phase. Here students are given the opportunity to identify a context that provides global challenges they are passionate about addressing. Infectious diseases are often caused by pathogenic microorganisms, such as bacteria, viruses, parasites or fungi; the diseases can be spread, directly or indirectly, from one person to another. The emergence of these infectious diseases is thought to be driven largely by socio-economic, environmental and ecological factors. Infectious diseases can be spread through direct transmission when disease-causing microorganisms are passed from the infected person to the healthy person via direct physical contact, and this often happens through blood or body fluids. Infections can also be spread through indirect transmission via sneezing or coughing. Next, let us try to develop a background theory for understanding how to describe the dynamics of transmission of infectious diseases.

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2.2 The Theorize Phase In order to theorize a model, we will need to make suitable assumptions that will help us lead to a mathematical model. For example, in order to develop a good theory to describe disease transmission, we must teach students about how diseases spread. Understanding the dynamics of infectious diseases using mathematical models can be traced back to first epidemic models proposed by Kermack and McKendrick in 1927 [17]. This original compartmental model assumed that the total population N may be divided into three distinct classes of sub-populations S, I and R denoting the susceptible, infected and the recovered classes, respectively. The susceptible class of individuals includes members of the population who have the potential to contract a disease and their size is denoted by S. The infected class of individuals are those who are assumed to have contracted the disease and this class is denoted by I . The final class of individuals denoted by R consists of those who recovered and cannot contract the disease again. Further, it is also assumed that the number of individuals in each of these classes (compartments) change with time, that is, S(t), I (t) and R(t) are functions of time t and the total population N is the sum of the number of individuals in these compartments. Hence, N = S(t)+I (t)+R(t). Since models represent approximations to reality, one must make suitable assumptions to simplify reality. In the development of the mathematical model, two principal assumptions were made in the Kermack–McKendrick model that included that infected individuals are also infectious and that the total population size N remains constant. This simple SIR epidemic model can be illustrated in compartments as in Fig. 2. Note that β and α are the transmission and recovery rates, respectively. It is assumed here that each class resides within exactly one compartment and can move from one compartment to another. Each compartment in the figure is represented by a box indexed by the name of the class, and arrows denote the direction of movement of individuals between the classes. When a susceptible individual enters into contact with an infectious individual, then that susceptible individual becomes infected with a certain probability and moves from the susceptible to the infected class. Therefore, the susceptible population decreases in unit of time by all individuals who become infected in that time while at the same time the class of infectives will be assumed to increase by the same number of newly infected individuals.

2.3 The Formulate Phase With this background theory, one can then accomplish the third phase of mathematical modeling, which is formulate. Specifically the dynamics of the three Fig. 2 Compartmental model for SIR model

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sub-populations S(t), I (t) and R(t) may be then formulated using the following SIR model given by first-order coupled ordinary differential equations (ODE): dS = −β S I dt dI = β S I −α I dt dR =αI dt

(1)

Note that this closed system does not include any births/deaths, for simplicity. Note also that in the first equation the rate at which susceptible members of the population become infected is proportional to the number of interactions there are between members of the population, and this is captured via the transmission rate β. The second equation indicates that each member lost from S(t) moves to population I (t) and that members of I (t) recover at a certain rate α. Rates β and α are often associated with the nature of the specific disease and typically found experimentally.

2.4 The Describe Phase This SIR model in system (1) is fully specified once we describe the transmission (β) and recovery (α) rates along with a set of initial conditions S(0), I (0) and R(0). The total population N at time t = 0 is given by N = S(0) + I (0) + R(0). Adding all the equations in system (1), we notice that N  (t) = 0 and therefore N (t) is a constant and equal to its initial value. One can further assume R(0) = 0 since no one has yet had a chance to recover. Thus a choice of I (0) = I0 is enough to define the system at t = 0 since then S0 = N − I0 . If the epidemic is triggered by a single infected individual, one can take t = 0 to be the moment at which I = I0 = 1. Note that in system (1), βSI denotes the number of individuals who become infected per unit of time. It is also assumed that the individuals who recover or die leave the infected class at a constant per capita probability per unit of time, which is the recovery rate α. This implies that αI is the number of individuals per unit of time who recover and therefore leave the infectious class and move to the recovered class.

2.5 The Analyse Phase Now, one can analyse the system (1) to make some important observations. First, note that S = βα makes the second equation in system (1) to be zero, namely, I  (t) = 0. Since the derivative is zero, the function I (t) can have a stationary

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(critical) point at some time. On the other hand, the number of infected individuals may be monotonically decreasing to zero or may have non-monotone behaviour by first increasing to some maximum level, and then decreasing to zero. It can also be noted that the spread starts to increase if I  (0) > 0, which yields the following necessary and sufficient condition for an initial increase in the number of infectives given by: β S(0) > 1. Thus if S0 < βα , the infection dies out and there is no α epidemic. The ratio of βα is referred to as the basic reproduction number, which denotes the number of secondary infections generated by an infected human when the population being considered is comprised of primarily susceptible humans.

2.6 The Simulate Phase The next step in mathematical modeling is to apply the model developed to real data. Consider the following popular dataset that appeared in the British Medical Journal Lancet reported on March 4, 1978, about an outbreak of influenza virus in a boys boarding school [3]. The school had a population of 763 boys, all of whom were at risk during the epidemic. One boy who had returned from an overseas trip is believed to have initiated an influenza epidemic in the school after his return. At the outbreak of the epidemic, none of the boys previously had influenza and so no resistance to infection was present. Of these 512 were confined to bed during the epidemic, which lasted from January 22, 1978, until February 4, 1978. Consider the related representative dataset illustrated in Table 1. Based on what one assumes is known in the SIR model in system (1), one can next simulate a variety of useful quantities. In order to do that one must have estimates for the parameters β and α. Since the epidemic was initiated by one sick boy infecting Table 1 Sample influenza dataset [3]

Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Infected number 3 8 28 75 221 291 255 235 190 126 70 28 12 5

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Fig. 3 Dynamics of SIR system (1) in comparison to data for α = 0.5 and β = 0.0026

two more sick boys 1 day later, a crude approximation could be S  (t) ≈ −2 per individual per day. With S0 = 762 and I0 = 1, one can estimate the initial  2 transmission rate to be: β ≈ −SS I(t) ≈ 762×1 = 0.0026. The report indicated that the boys were taken to the infirmary within 1 or 2 days of becoming sick. So one may estimate that 50% of the infected population was removed each day, or α = 0.5 per day. The dynamics of the model system (1) solved and simulated numerically using Runge–Kutta methods in comparison to the data is shown in Fig. 3. Note that we use the MATLAB software and the built-in ode45 that implements fourth-order Runge–Kutta scheme to solve the differential equation system.

2.7 The Validate Phase Note that we had used a crude calculation to estimate the parameters β and α, which helped us to simulate the dynamics of system (1). However, it is important to validate our model and computations as well as estimate optimal parameters for the given data. One way to efficiently do this is to employ techniques that search for a local minimum using a least-squares regression type approach. In order to do this, we create an objective function represented by squared differences in the daily infected counts from observed data and the computer simulated data. This objective function is then minimized for the initial conditions S0 = 762, I0 = 1, R0 = 0

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Fig. 4 Dynamics of the SIR system (1) in comparison to data for α = 0.4768 and β = 0.002568

starting with poor guesses for αguess = 0.1, βguess = 0.01. We take advantage of MATLAB’s fminsearch function to estimate the optimal parameters to be β = 0.002568 and α = 0.4768. The predicted dynamics of the three subpopulations for these optimal parameter values along with comparison to the true data is illustrated in Fig. 4.

2.8 The Predict Phase Finally, to study the predictive capability of the model, one can employ physicsinformed deep learning models [23–25] to leverage the hidden physics of infectious diseases and infer the latent quantities of interest (i.e., S, I and R) by approximating them using deep neural networks [26]. Such physics-informed neural networks differ from the standard deep learning models as they include neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general differential equations that model the system. Specifically, the three sub-populations can be approximated using a backpropagation algorithm [15], and the parameters α and β of the differential equations turn into parameters of the resulting physics-informed neural networks. The results are reported in Fig. 5 where the algorithm learns the parameters α and β to have values 0.454864 and 0.00229365, respectively. Predictions of the model for S, I and R are plotted using solid blue lines, while the data are depicted using red

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500

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Fig. 5 Physics-informed deep learning model predictions and comparison to data with parameters α = 0.454864 and β = 0.00229365

asterisks [26]. Note that the values of α and beta obtained via the physics-informed approach are close to the values obtained via the fminsearch used in the validate phase. While the fminsearch employs a direct search method (Nelder–Mead Simplex), it is very sensitive to initial conditions. The deep learning on the other hand offers multiple options with choice of layers; the type of neural network, the propagation algorithm, weights and all these can help with better performance than just a traditional optimization algorithm.

3 Global Problem Solving Using Mathematical Modeling Now that we have established the importance of how to interpret the different phases of the mathematical modeling cycle illustrated in Fig. 1, it is easy to note that this cycle could be applied to other complex physical problems. In this work, we had considered the SIR system (1) describing the spread of infectious diseases to help students learn about the mathematical modeling cycle (Fig. 1), for simplicity. We are aware that most diseases would require more sophisticated models beyond just the SIR system (1). One such example could include modeling vector-borne diseases such as malaria where one must include a latent phase during which the individual is infected but not yet infectious. This delay between the becoming infected and the infectious state can be included in the SIR model by adding a exposed subpopulation, E, and letting infected (but not yet infectious) individuals move from the S compartment to E compartment and from E compartment to I compartment. Next, we will build on what we have learnt so far to develop a model to prevent the spread of the Zika virus. Global Challenge: Ever since the Zika virus outbreak in the Americas in 2015, several countries have reported the spread of the infection. It is now known that the transmission of Zika virus can both be spread by the Aedes aegypti mosquitoes and sexual contacts with infected and non-infected persons. The global challenge to solve is to identify effective ways to control the Zika virus transmission.

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3.1 The Observe Phase The Zika virus (ZikV) that is believed to be transmitted by the infected Aedes aegypti mosquitoes has been named as a global health threat [11]. There are several factors that make this disease unique including the fact that the virus can be spread through both vector-borne transmission and sexual transmission. Here the vector refers to the mosquito that transmits the disease from infected humans to a healthy human. Because of the absence of efficacious drugs and/or vaccines, the World Health Organization’s 2018 annual review declared that there is an urgent need for accelerated research and development for ZikV disease [36]. Vector control measures are of fundamental importance in the fight against Zika. Over the last two decades, some prominent approaches for controlling vector populations, recommended by the World Health Organization (WHO) and the Centers for Disease Control and Prevention (CDC), involve the use of insecticidetreated mosquito nets (ITN) and Indoor Residual Spraying (IRS) [21]. Using ITN can help reduce contacts between mosquitoes and humans at home. Further, mosquitoes that remain within the boundaries of sprayed homes after their meals can die as a result of IRS. Moreover, the American College of Obstetricians and Gynecologists (ACOG) recommended the use of condoms to prevent spread of Zika through sexual transmission. It is therefore important to study the impact of using selected preventive measures such as ITN, IRS or condoms in controlling or ameliorating the spread of the Zika virus and makes an indirect effort to assess the potential contributions of sexually transmitted Zika infections.

3.2 The Theorize Phase The transmission dynamics may be described via compartmental models describing the rates of different human sub-populations including Susceptible, Infected and Recovered (SIR). Since ZikV transmission is closely related to dengue, and yellow fever viruses, it can be modeled with similar SEIR–SEI models [7, 8]. Note that the introduction of the E compartment is needed to reflect the latency period. Specifically both the humans and the vectors who are exposed (E) have had contact with an infected person but are not themselves infectious. While these models provide good insight into the interaction of the humans and the infected mosquitoes during ZikV transmission, they do not include the life stages of the major vector of ZikV. Also, the use of condoms as an effective preventative strategy has not been explored greatly in ZikV transmission models. In this chapter we make an attempt to enhance a mathematical model of ZikV transmission introduced in [18] with new preventative measures such as inclusion of ITN, IRS and condoms as additional preventative strategies.

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3.3 The Formulate Phase Let us consider a ZikV model based on SEIR host model, SEI vector model, using logistic growth in human and vector population and added vector life stages. The compartments for humans, subscripts with H; and compartments for vectors, subscripts with V. Let S(t), E(t), I (t) and R(t) denote the number of susceptible, exposed, infected and recovered individuals at time t. The human population at time t is given as NH (t)=SH (t)+EH (t)+IH (t)+RH (t). The vector population at time t is given as NV (t)=SV (t)+EV (t)+IV (t). For vector compartments, let V1 (t), V2 (t) and V3 (t) denote the number of eggs, larvae and pupae of vectors in the environment, respectively. We assume that there are no infectious compartments for these three stages, since it is documented that only female adults bite humans and transmit the virus. The system of coupled SEIR–SEI ordinary differential equations describing the interactions in the compartmental model (Fig. 6) may be then expressed as SH IV IH SH − α(1 − C) ) − βb(1 − I T N ) SH S˙H = λNH (1 − KH NH + M NH E˙H = βb(1 − I T N )

IV IH EH SH + α(1 − C) SH − νH EH − λNH NH + M NH KH

˙ = νH EH − γH IH − λNH IH

IH KH

RH R˙H = γH IH − λNH KH V˙1 = eV NV − τ1 V1 − μ1 V1 − j · I RS · V1

(2)

V2 V˙2 = τ1 V1 − τ2 V2 − μ2 V2 − κ 2 − j · I RS · V2 Kv V˙3 = τ2 V2 − τ3 V3 − μ3 V3 − j · I RS · V3 S˙V = ρτ3 V3 − βb(1 − I T N )SV E˙V = βb(1 − I T N )SV

IH − μV SV − h · I T N · Sv − j · I RS · Sv NH + M

IH − μV EV − νV EV − h · I T N · Ev − j · I RS · Ev NH + M

I˙V = νV EV − μV IV − h · I T N · Iv − j · I RS · Iv

Fig. 6 Compartmental model where bold lines refer to transmission within each population and dotted lines refer to transmission between human–vector and human–human population

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The first four equations describe the dynamics of each of the compartments of the humans. During the outbreak, the human population will be assumed to follow a logistic growth with human population growth rate λ and carrying capacity KH , and the limited carrying capacity is distributed in four compartmental models. Such a logistic term can be seen in the terms on the right-hand side with KH as the denominator. The susceptible population SH (t) will move to the exposed class EH (t) after acquiring the disease through infected vector and sexual transmission. Let b denote the average biting rate per unit time of a female vector, β represents the percentage of bites that transmit ZikV, M is the amount of alternative hosts besides human in the environment and α denotes the sexual transmission rate of ZikV. These are represented by the last two terms on the right-hand side of the first equation and the first and the second term in the second equation. Note that the exposed class EH (t) is the population of human incubation period, and the inverse incubation time of ZikV for human is assumed to be νH . Infectious class is assumed to recover with a rates of γH . This is seen in the term νH EH in the second and third equations and γH in the third and fourth equations. For the mosquito, let eV denote the number of eggs laid per female vector per unit time, τi and μi , i ∈ {1, 2, 3} be the reciprocal of the development time and mortality rate in stages i, respectively. Notice that larvae live in water. With the limited resource, the larvae population will be expected to follow a logistic growth with carrying capacity Kv and mortality rate caused by limited carrying capacity κ. Let ρ be the percentage of females among adult vectors. The individual susceptible vectors SV (t) move to the exposed class through biting an infected human and natural mortality. The mortality rate is given by μ. We will also assume that vectors of exposed class move to infectious with a vector incubation rate νV . For computational experiments that will be done in the later simulate phase, we will use the parameter values described in Tables 2 and 3.

3.4 The Describe Phase To enhance the mathematical model in system (2), we incorporate various preventative measures. The effect of condoms C is introduced in the rates of sexual transmission from the susceptible human class to exposed human class through a parameter measured as a percent (1 − C). If C = 0, humans do not use condom, which will potentially lead to sexual transmission causing ZikV transmission; if C = 1, no ZikV transmission occurs through sexual contact. Similarly we also incorporate the effect of insecticide-treated nets as ITN. The effect of ITN is introduced in the rates of transmission of vector bites transmission from the susceptible human class to the exposed human class through (1 − I T N ). If ITN=0, the nets have no effect, disease spread through vector transmission; if ITN=1, the only movement from susceptible to exposed human class is via sexual contact. The parameters for the removal of mosquitoes were denoted by h and

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Table 2 Parameters’ definitions and references Name Number of infectious bites delivered per vector per day Human incubation rate Human recovery rate Natural death rate for vector Vector incubation rate Parameter for ITN rate Parameter for IRS rate Parameter for C rate Number of eggs laid per female mosquito per unit time Egg development rate Larva development rate Pupa development rate Egg mortality Larva mortality Pupa mortality Female adult percentage

Notation βb νH γH μ νV h j α eV τ1 τ2 τ3 μ1 μ2 μ3 q

Value 0.07 1/3 1/9 0.05 0.1 1/365 1/365 0.2 80 per day 3 days 9 days 3 days 0.05 0.025 0.0025 0.68

Reference [7, 8] [7, 8] [7, 8] [8] [8] [7] [7] [8] [8] [8] [8] [8] [8] [8] [8] [8]

Table 3 Estimated parameter values used in the simulation Name Initial human population Initial human infections Initial adult vector population Initial exposed adult vectors Initial infectious adult vectors Initial number of eggs Initial number of larva Initial number of pupa Carrying capacity for human Carrying capacity for larva Larva mortality rate due to limited carrying capacity Number of alternative hosts Human population growth rate

Notation NH (0) IH (0) NV (0) EV (0) IV (0) V1 (0) V2 (0) V3 (0) KH KH κ M λ

Rio de Janeiro State 16,000,001 1 70,000,000 5 5 180,000,000 90,000,000 70,000,000 30,000,000 100,000,000 1 100,000 0.00005

j associated, respectively, with ITN and IRS. To account for a wide range of behaviours, one can let the values of ITN, IRS and C to range from 0 to 1.

3.5 The Analyse Phase The basic reproduction number, denoted by R0 [7, 8], can be used to determine whether there is an outbreak or not. In order to derive it for the system (2), we use

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the Next Generation Matrix approach [6] to derive the R0 for system (2). In this method, two important quantities are calculated including the rate of appearance of new infections in any compartment and the rate of transfer of individuals coming into a compartment and moving out from it. Using these, one computes the Next Generation Matrix and R0 may then be computed as the largest eigenvalue of this matrix. Theorem 1 The basic reproduction number R0 for system (2) is given by R0 =

νH α(1 − C) 1√   + R NH NH 2 2 νH + λ γH + λ KH KH 

where R = R1 + 4R2 with 2 α 2 (1 − C)2 νH    NH 2 NH 2 νH + λ γH + λ KH KH

R1 = 

⎤ NH NV 2 (1 − I T N)2 ν ν (βb) H V ⎥ ⎢ (NH + M)2 ⎥     R2 = ⎢ ⎦ ⎣ NH NH νH + λ γH + λ (μV + νV + P ) (μV + P ) KH KH ⎡

Proof Given the infectious states EH , EV , IH and IV in system (2), we first create a vector F that represents the new infections flowing only into the exposed compartments given by

F = βb(1 − I T N)NH



IV IH + α(1 − C)IH , βb(1 − I T N)NV , 0, 0 NH + M NH + M

Along with F , we will also consider V, which denotes the outflow from the infectious compartments in the equations in system (2) by

 EH IH V = νH EH + λNH , (μV + νV + P )EV , −νH EH + γH IH + λNH , −νV EV + μV IV + P IV KH KH

where P = (h · I T N + j · I RS). Next, we compute the Jacobian F from F as ⎞ ⎛ H 00 α(1 − C) βb(1 − I T N ) NHN+M ⎟ ⎜ V 0 ⎟ ⎜0 0 βb(1 − I T N ) NHN+M F=⎜ ⎟ ⎠ ⎝0 0 0 0 00 0 0

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and the Jacobian V from V is given by ⎛

NH 0 νH + λ K H ⎜ 0 μ + ν ⎜ V V +P V=⎜ ⎝ −νH 0 γH 0 −νV

⎞ 0 0 ⎟ 0 0 ⎟ ⎟. NH + λK 0 ⎠ H 0 μV + P

Using matrices F and V, one can compute the Next Generation Matrix FV−1 and calculate eigenvalues of FV−1 using the characteristic equation det (FV−1 − ΛI) = 0, where Λ denotes the eigenvalues of the matrix and I represents the identity matrix. The characteristic polynomial is, therefore, the following quadratic equation given by  νH + λ

NH KH



  NH (μV + νV + P ) γH + λ (μV + P )Λ2 KH +(μV + νV + P )(−νH )(μV + P )α(1 − C)Λ − νV νH βb(1 − I T N)(

NV NH )( )=0 NH + M NH + M

The proof is completed by computing the basic reproduction number R0 , which corresponds to the dominant eigenvalue given by the root of this quadratic equation.

3.6 The Simulate Phase In this section, we implement and simulate the solution to the system (2) in MATLAB using fourth-order Runge–Kutta method. For our simulations, we employ parameters summarized in Table 2 that are used in modeling the ZikV disease dynamics from various references. Table 3 summarizes the estimated parameter values used in the simulation of ZikV disease dynamics for Rio de Janeiro State, Brazil, between January 2015 and July 2015. Also, for our simulations, we choose the initial populations to be SH (0) = 16, 000, 000, EH (0) = 0, IH (0) = 1, RH (0) = 0, SV (0) = 69, 999, 990, EV (0) = 5, IV (0) = 5, V1 (0) = 180, 000, 000, V2 (0) = 90, 000, 000 and V3 (0) = 70, 000, 000. The dynamics of all sub-populations for both human and adult vector with no protection are shown in Fig. 7. For the first 200 days, human infectious population reaches the peak on day 98, 4,341,000 individuals, 27% of the total populations, catch ZikV. The reproduction number R0 = 2.0327.

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Fig. 7 Dynamics of human and adult vector sub-populations

3.7 The Validate Phase Figure 8 contrasts the containment effectiveness for different strategies. These include increasing insecticide-treated mosquito nets usage rate (ITN) only, increasing indoor residual spraying usage rate (IRS) only, increasing condom usage rate (C) only and increase of all three preventative strategies. We increase each rate in 20% intervals. Figure 8a shows that increasing ITN usage only by 20% will lead to reduction of 13% of total infectious population (3,791,000 infectious individuals). Increasing the indoor residual spraying usage rate alone does not seem as effective as ITN. Figure 8b shows that increasing IRS rate by 80% will lead to only a reduction of 101,000 individuals to get infection (2.3% reduction). Nonetheless, spraying is one of the important and common strategies for mosquito control. Similarly Fig. 8c shows that increasing condom usage rate of human was seen to be also only slightly effective. Figure 8c displays that increasing condom rate by 60% will lead to reduce 5.4% of infectious population. Figure 8d shows that the combination of the strategies is the most effective protection that helps to validate our model. Increasing ITN, IRS and C rate by 20% will reduce 676,000 individuals to catch ZikV (16% reduction). Similar observations were made for the dynamics of the adult vector populations. The influence of ITN (which seems to have the most effect) and IRS (which is less effective) is illustrated in Fig. 9. Figure 10 illustrates the dynamics of eggs, larvae and pupae with IRS set to 20%, and the graph shows a typical logistic behaviour for all three populations. We noted that as the IRS rate increased, the height of the logistic graphs decreased.

3.8 The Predict Phase Now, that we have a model that has been validated, the next step is to use it to predict what effective preventative measures can help prevent outbreaks. This

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Fig. 8 Dynamics of infectious humans for three containment and combination strategies

Fig. 9 Dynamics of infectious adult vectors by increasing ITN and IRS by 20% each time

can be predicted using the basic reproduction number R0 . Figure 11 displays the reproductive number R0 as a function of ITN, IRS and C. Notice that as the preventative measures increase, the R0 decreases. Figure 11a–c also gives evidence that ITN is the most effective prevention among these three strategies. However, when ITN=1, R0 =1.1997 demonstrates that the influence of ITN is still not enough

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Fig. 10 Dynamics of eggs, larvae and pupae for IRS = 20%

Fig. 11 Basic reproduction number for three containment strategies and combination strategies

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alone to bring R0 < 1 and prevent an outbreak. The combination strategies used in Fig. 11d shows that an ITN increase of 60%, IRS and C increase of 40% will lead to R0 < 1, meaning that the infection will die out in the long run.

4 Integrating Research with Teaching and Learning While the focus of this paper is on using infectious diseases as a problem context to engage students in applied mathematics education research through the eight phases, each of these proposed phases also provides the educators an opportunity to enhance their pedagogical practices in teaching this material to increase student learning. Specifically we use this opportunity to engage the students to become aware of the SDG2030 goals and in particular learn about SDG-3. We can have them observe and report on how this SDG-3 has been classified into 13 Targets that specify the goal and learn about the 28 Indicators that represent the metrics by which the world aims to track whether these targets are achieved. These can be used by the students to connect with local contexts of the problem. One way to engage students to theorize is to motivate the nature of infectious diseases modeled via compartments. This can be done through simple activelearning strategies in the classroom. For example, one can create a simple SIR game that includes everyone one in the class who are given a paper with a number written on it. This secret number is only known to each individual. Only one of the individuals has the number zero at the beginning of the game to denote that they are infected. The game runs in several rounds, and in each round, students are asked to meet a different partner they have not met before, shake hands (as a way of making direct contact) and multiply the two numbers on their sheet and update their respective sheets with this newly computed product. Clearly as the rounds proceed, more members start to become infected as they meet individuals who have the number zero. These results can then be finally formulated in the form of tables and graphs to illustrate how the dynamics of the SIR model evolves as the game proceeds. Such simple ways to engage students in learning are very powerful to help them understand the concepts. The graphs discussed through the mutual interaction can then be used with ideas of slope fields they may already be aware of and describe related rate of change using differential equations. The students can also be encouraged to perform simulations by recording the data on simple platforms such as EXCEL and perform related analysis and simulations. For helping the students understand validation, they can learn to work with real datasets of similar diseases that are publicly available from websites or journal articles and have them verify the trends and behaviours and study the reliability of their model. They can also use this opportunity to make suitable predictions on parameter values that can try to reproduce given datasets. Note that there are opportunities along the way for students in this active learning experience to exercise their twenty-first century skills including communication,

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collaboration, creativity and critical thinking as well as experience important competencies needed in global problem solving such as failure, fluency and flexibility.

5 Discussion and Conclusion In this chapter, we present some potential ways to engage students in applied mathematics education and research. First, to help students identify a problem, the chapter helps readers become aware of the United Nations SDG2030 framework, which provides a natural context for identifying potential global challenges. The chapter then demonstrates how to solve these challenges using a new mathematical modeling cycle for global problem solving. After introducing the various phases in the modeling cycle for a simple SIR infectious disease model, the cycle was applied to a more complex global problem of controlling the spread of the Zika virus. Specifically, the mathematical modeling cycle examined the effectiveness of three containment strategies: increasing insecticide-treated nets usage rate, increasing indoor residual spraying usage rate and increasing condom usage rate. Through computational simulation of real data from Rio de Janeiro State, we were able to further investigate and find that ITN is the most effective preventive measure. Our study gives an insight for control, mitigation and elimination of the spread of ZikV disease and can help decision maker to select and invest the best combination to fight the spread of infection. We hope that the presentation helped to provide insights to readers on identifying a context for global problem solving from SDG2030, which can then be combined with a mathematical modeling cycle to provide a rigorous content to help engage students in applied mathematics education and research.

References 1. Albrecht, J.R., Karabenick, S.A.: Relevance for learning and motivation in education. J. Exp. Educ. 86(1), 1–10 (2018). DOI: 10.1080/00220973.2017.1380593 2. Anderson, R.M., May, R.M.: Population biology of infectious diseases: Part I. Nature 280(5721), 361 (1979) 3. Anonymous: Influenza in a boarding school. Br. Med. J. 1, 578 (1978). http://www.ncbi.nlm. nih.gov/pmc/articles/PMC1603269/pdf/brmedj00115-0064.pdf 4. Bliss, K.M., Galluzzo, B.J., Kavanagh, K.R., Skufca, J.D.: Incorporating Mathematical Modeling into the Undergraduate Curriculum: What the GAIMME Report Offers Faculty. PRIMUS (2018), pp. 1–29 5. Boaler, J.: The role of contexts in the mathematics classroom: do they make mathematics more “ Real”?. Learn. Math. 13(2), 12–17 (1993) 6. Brauer, F., Castillo-Chavez, C.: Mathematical Models in Population Biology and Epidemiology, vol. 40. Springer, New York (2012) 7. Brauer, F., Castillo-Chavez, C.: Mathematical Models for Communicable Diseases, vol. 84. SIAM, Philadelphia (2012)

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Developing Non-Calculus Service Courses That Showcase the Applicability of Mathematics Lucas Castle

Abstract Students often take precalculus or college algebra as a terminal math course, leaving them with the impression that mathematics lacks real meaning. As applied mathematicians, we are well-poised to intervene and design inspiring general education courses that reveal the utility of mathematics. In this paper, we share experiences working with faculty from a range of disciplines to develop a two-course sequence that explores mathematical concepts by answering questions that matter to our student population.

1 Introduction An increasingly interdisciplinary world demands strong quantitative reasoning and problem-solving ability [1, 2]. Colleges must encourage the development of these essential skills in students by demonstrating the involvement of mathematics in fields such as biology, social sciences, business, finance, computer and information systems, etc. [3]. However, many students satisfy their major’s mathematical requirement through courses in college algebra, precalculus, or a simplified discipline-specific calculus. A significant problem is that these general education math courses tend to lack sufficient meaningful context. An endless supply of reductionist examples and “toy” problems leave students feeling uninspired and without any sense of the utility of mathematics. Preparing students in service courses for the quantitative challenges in their lives and careers requires contextualized, open-ended problems based on those they will likely face in the real-world. This quantitative reasoning is vital regardless of field of study, and should be pursued from a disciplinary and application-focused perspective [4–6]. Furthermore, contextualized material yields a more engaging classroom environment where students obtain deeper understanding of the material [7–9].

L. Castle () North Carolina State University, Raleigh, NC, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 R. Buckmire, J. M. Libertini (eds.), Improving Applied Mathematics Education, SEMA SIMAI Springer Series 7, https://doi.org/10.1007/978-3-030-61717-2_4

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Applied mathematicians are well-poised for the challenge of developing and teaching service courses that develop quantitative literacy through relevant mathematics. Their expertise in a wide array of fields, including industry, business, medicine, and engineering, paves the way for the design of curricula that targets the specific audience of lower-level service courses, activities that strengthen non-mathematical “soft-skills” (professional presentations, written reports, basic computational tools, for example), and classroom experiences that connect multiple disciplines and ways of thinking. In this article, we discuss the implementation of an introductory level mathematics course sequence designed by applied mathematicians in conjunction with faculty members across six other departments at Virginia Military Institute. These courses, which are built upon a modeling foundation, contain a series of contextspecific quantitative reasoning modules and activities. We discuss how to implement an application-centered approach to non-calculus math classes, provide several examples of activities developed by cross-disciplinary teams for these courses, and provide preliminary feedback from students from the first two cohorts of the courses.

2 Implementation Virginia Military Institute is a state funded military college with a liberal arts focus. The student body consists of around 1700 undergraduates who live on campus and attend classes full-time. Of these students, 61% come from the state of Virginia, and roughly 11% are females. Every student, or cadet, must participate in the Reserve Officers’ Training Corps (ROTC) program at VMI in order to complete their degree, with roughly 50% commissioning with the military upon graduation. The remaining cadets pursue careers in government, industry, or attend graduate school. Additionally, a significant portion of a cadet’s academic week consists of physical training and other institute-specific activities, such as inspections or military instruction. Thus VMI has a unique and challenging learning atmosphere, as cadets cannot devote 100% of their attention to the classroom. In response to both the increasing need for a greater multidisciplinary focus in mathematics and an engaging, relevant curriculum, VMI has embraced teaching applied mathematics to all students. This initiative led to the creation of Math that Matters, a two-semester introduction to statistics, data analysis, and problem solving for students whose degree plans do not list calculus as a required course. Every topic, tool, and problem covered in the course is presented from a math modeling, project-focused perspective. Practical problems motivate the mathematical content, and students use mathematical tools to explore and develop understanding of problems in the real-world [10, 11]. Math modeling also provides a platform to develop non-mathematical skills frequently cited as desirable by the workforce. In fact, employers indicate that the skills they need must include proficiency in oral and written communication, project management and leadership, teamwork and interpersonal ability, and

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problem-solving skills [12–14]. By focusing on applications, students have ample opportunities to practice effective communication in a variety of mediums, such as writing, data visualization, and presentations. The difficult yet relevant nature of open-ended, real-world problems encourages working in teams which exposes students to multiple perspectives and ways of thinking. Furthermore, group work promotes initiative and leadership as students delegate tasks and fulfill roles towards the completion of projects. Generally speaking, the goals of Math that Matters include empowering students with mathematical concepts relevant to their disciplines and future careers, developing soft-skills (through presentations, group work, written assignments, and technology), promoting attitudinal shifts toward mathematics that support learning, and providing an engaging environment for learning mathematics collaboratively. Application projects, which were co-developed with the Biology, Modern Languages, History, International Studies, Physical Education departments, and Economics and Business, supply the necessary context to generate interest and investment in the students. We dive into the incorporation of these endeavors by discussing in detail the activities developed through cross-disciplinary collaborations. Connecting students with mathematics in the classroom involves finding out what problems matter to them. This design philosophy not only reveals the utility of mathematics to those who usually dismiss it outright but also serves to garner buy-in for increased engagement in the classroom. Applied mathematicians are uniquely positioned to connect students to mathematics in the real-world. Training and experience serve as leverage for designing meaningful activities and assignments, teaching the use of mathematical tools to solve real-world problems, creating interdisciplinary experiences inside and outside of the classroom, and empowering students to encounter mathematical ideas across disciplines without fear or hesitation. When choosing topics for Math that Matters, we considered the population our courses serve and methodically created assignments and projects throughout the design process. The following key ideas summarize this process: • What sorts of problems will arise in the future for students who continue in careers associated with these disciplines? Identify the majors of the students, and consider the problems associated with those majors that contain quantitative elements or that benefit from a problem-solving (i.e. math modeling) approach. • What tools, analytic and technological, would bolster students’ future success in the workforce? Include skills that students need in order to pursue success in their chosen majors and careers. Create assignments that encourage the development and practice of soft-skills, such as creating presentations in PowerPoint or documenting results (through reports, memos, etc.). • Who are the colleagues that can work with you in the design of your course? Collaborate with faculty and professionals in the fields of interest to the target audience of your course. Take advantage of the expertise of those who have a

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deeper understanding of the discipline in order to best frame the problems with quantitative elements within that field of study. We structured the Math that Matters courses as follows. The first semester serves as an introduction to data analysis, including topics such as communicating data, descriptive statistics, inferential statistics (confidence intervals and hypothesis testing), and curve-fitting. Students bring their computers to class daily, and they learn Excel from the ground up as a computational tool for solving problems. Various types of presentations, writing assignments, and projects help prepare students for the open-ended problem they face in the following semester of the course sequence. The second semester provides examples of the math modeling process through applications including art gallery design, creating metrics and statistical studies, and conducting sensitivity analysis on retirement plans. Students also choose a topic of interest, apply mathematical tools learned throughout Math that Matters to answer a question related to that topic, and present their results at a course-wide poster session at the end of the semester. The courses initially launched during the 2018–2019 academic year, and around 300 students enrolled in the sequence across 12–14 concurrent sections a semester. It is important to note that class sizes capped out at around 22 students, allowing for the implementation of instructor-facilitated collaborative learning experiences. These courses use a student-centered approach where the classroom serves as a forum in which instructors guide learning and encourage peer-to-peer discussion through inquiry. Classes were taught by full-time faculty and adjuncts, some of whom were not directly involved in the creation of course content. (One of the adjuncts helped design the module on descriptive statistics, the other adjunct was only involved in teaching the course.) That said, all faculty involved in teaching the class attended weekly meetings, giving insights into the effectiveness of various modules and teaching strategies that often led to revisions of the class materials.

3 Example Modules and Activities In this section, we present examples of activities and modules developed in conjunction with non-mathematics faculty at Virginia Military Institute for the Math that Matters course sequence. Table 1 below provides summarizes materials produced and their respective partner discipline. In Sects. 3.1–3.6, we highlight three activities from the first semester (Culture Trip, BATtleship, and Voting Tendencies) and three activities from the second semester of the sequence (Art Gallery, Leaking Pool, and Army Finance). Each of these incorporates a significant contribution from a partner discipline and represents a large component of an in-class module.

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Table 1 Math that Matters Materials created with partner disciplines Module/Activity Culture Trip (Intro to Spreadsheets) Fitness Scores (Descriptive/Inferential Statistics) BATtleship (Intro to Sampling) Voting Tendencies (Hypothesis Testing) Local Historical Population Trends Art Gallery (Intro to Modeling) Good Cadet (Statistical Study Design) Leaking Pool (Math Modeling) Cobb-Douglas Model (Math Modeling) Army Finance (Sensitivity Analysis)

Partner discipline Modern Languages Athletics Biology International Studies History English and Humanities Commandant’s Staff Athletics Economics and Business Army ROTC

3.1 Culture Trip The Culture Trip module, developed in conjunction with the Department of Modern Languages and Cultures, is one of the first activities students engage in during the course sequence. In this module, students must plan a trip of cultural value to a foreign country. Constraints for the problem include choosing a country in which the currency conversion is not 1:1 with the US dollar and the trip must span roughly 3 days. As they research their itineraries in groups of 2–3, they record their budget for the trip in a spreadsheet, learning how to input data and create a table in Excel. Ultimately, students write a proposal presenting their budgets, justifying the educational value of the trip and requesting funds from a potential donor. On the first day of the module, students form groups and the instructor introduces roles: recorder, reporter, and facilitator. Assignment of groups and roles are left to instructor discretion, with the caveat that individual students experience different roles throughout the course. Strategies used by instructors for selecting groups included self-selection, using playing cards (i.e., all aces form a group) or other forms of random assignment, and intentional pairings of students (for various academic and interpersonal reasons). The recorder’s responsibility is the material—any notes taken, any written work completed must be done and disseminated to the group by them. The reporter is the authorized representative of the group, and they will share the group’s ideas during class discussions. The facilitator works to ensure their group is on schedule and on task. They are also suggested to read handouts aloud to their group and encourage all team members to participate. After the students have selected or been assigned roles, they are given 20 min to conduct some preliminary research on their trips. The instructor prompts the class with questions that guide students in developing their budget, such as: What will you do to build a cultural experience? How much does each activity cost? Do you need to book or pay in advance? What other costs should you consider? Students are also told that, at the end of the 20 min, their reporter should be prepared to share a 2 min

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briefing on what their group plans to do and how they will spend money throughout the trip. The instructor then leads a class discussion, having students think about expenses they may not have considered based on what other groups shared. After the students reflect on their own and their peers’ trips, they then spend the remainder of class refining budgets and preparing a draft proposal. Students begin the next class day pinning hard copies of their proposals, including the budget they created in a spreadsheet, up around the classroom. They participate in a gallery walk, briefly leafing through proposals around the room (initially as individuals, and then again in their groups). The instructor then leads a class discussion, asking students what style features stood out among the proposal drafts around the room. Students drive this discussion forward; however, instructors use this as an opportunity to highlight what “right” looks like, i.e. well-formatted and organized tables, and proposals with a clear introduction, body, and conclusion. The remainder of class is devoted to a scaffolded worksheet on using Excel, including topics such as inputting data, relative and absolute referencing, entering equations, and creating graphs. Students maintain their roles as they work through the worksheet, checking in with the instructor when prompted. Ideally, the students take insights from the class discussion and tools learned through the worksheet to update their budgets and write a final version of their team’s proposal (which is due on the following class day). A rubric is provided, highlighting features such as clarity, presentation of the budget using Excel, and strong justification of budgeting decisions.

3.2 BATtleship The BATtleship activity, which has cadets predicting the number of bats in a region, introduces students to some of the issues surrounding the topic of sampling. The activity is based on the research of a faculty member in the Department of Biology, which involves collecting data to attempt to extrapolate the total bat population on several talus slopes in the area for conservation purposes. In the activity itself, students are divided into groups. One student serves as “slope keeper,” and they are given the exact number and location of bats on a 10×10 grid. The remaining members of the group are instructed to take samples of bats of varying sizes (10 and 25), without any guidance on how to take their sample, similar to how the game battleship is played. They call out a cell on a grid by using letters and numbers that correspond to rows and columns, and the slope keeper reveals the number of bats found in that location. One of the slopes sampled provides a simplification of a typical distribution of bats (i.e. small “pockets” of 1–2 bats spread out sporadically), while another models the mothering season. It is possible, during the mothering season, to have large numbers of bats concentrated in one area with sparse distribution over the remaining area of the slope. Groups that do not find the large pockets of bats via their sampling procedure likely underestimate the total population, while those that happen to find

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the large pockets of bats tend to give an overestimation. After obtaining results, the instructor leads a discussion on the importance of larger samples sizes, random sampling, and understanding the underlying context of the system prior to taking a sample. Students ultimately realize that the best time to sample in this scenario is likely outside of the mothering season (for accurate extrapolations of the total population). While there is no major deliverable for this activity, it serves as an engaging introduction to the Sampling module that ties mathematics learned by students to a faculty member outside of the department.

3.3 Voting Tendencies As part of a Hypothesis Testing module, students are tasked with investigating possible factors that influence whether people attend political rallies, work campaigns, and turn out to vote. This assignment, co-written by faculty in the Department of International Studies, involves using Excel to run several chi-squared tests to determine if there is sufficient statistical evidence suggesting these outcomes depend on various qualities (such as income, ideology, party association, education, age, and gender, to name a few). Working in groups, students choose a particular factor that interests them and prepare a short slideshow presentation of their results that will be shared with the class. A spreadsheet with information on 5914 voters is provided for students to analyze. In a prior module, students learned how to construct and use pivot tables in Excel, a skill they leverage here to easily pare down the data set to the specific factor and outcomes. More specifically, they construct several contingency tables (or two-way tables)—one for the actual counts and another for the expected counts associated with each of the outcomes of interest. To compute the expected frequencies associated with a particular outcome, students leverage probability skills acquired in a previous lesson. From there, students run their chi-squared tests using a function in Excel, obtain and interpret their obtained p-values, and clearly communicate their results to the class. Students are responsible for stating their testable questions, null and alternative hypotheses from both quantitative and qualitative perspectives, and contextualized conclusions based on the results of their hypothesis tests.

3.4 Art Gallery The Art Gallery module, initially inspired by an exhibit once designed by a faculty member in the Department of English, Rhetoric, and Humanistic Studies, occurs at the beginning of the second semester of Math that Matters. This set of activities spans several class days and involves students working in groups to design both a

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small art exhibit (i.e. a display case) and a full art gallery. The ultimate deliverable for this section of the course includes a gallery design paper in which students justify the arrangement of the artwork and people spacing with regards to traffic flow through the exhibit. Furthermore, they exercise creativity by creating a 3D scale model of their proposed art gallery that they share during an in-class presentation. The first class day of the module consists of a guided tour of a museum on campus, where students learn about the factors that go into exhibit design. They are provided with a list of questions to consider as they move through the museum, revealing many of the nuances and constraints associated with gallery design, and helping students narrow down the open-ended nature of the problem. • What story does the artwork tell? • Who will be looking at the artwork, and how does that affect the placement of art and associated nameplates? • Is the exhibit 2D or 3D? • How many people can fit in the space comfortably? • Can you direct the flow of people in the design? The assignments in this module intentionally build upon one another. Before diving into a full gallery, students first design a 2D glass case containing 7 pieces of art by Renaissance printmaker Albrecht Dürer. The instructor provides a list of the pieces of art, complete with images and dimensions, along with a downloadable file with images of the artwork. Students spend time in class researching the artwork, designing their case (layout and spacing of the artwork), and producing a scale model on a PowerPoint slide. The instructor interfaces with groups, helping with scaling calculations and asking about design choices (perhaps using terms like “assumptions” or “modeling” when describing decisions made by the group or the process of designing the case itself, respectively). The next class begins with the small exhibit presentations, where students show and justify their model display case. Instructors provide feedback on design choices and justifications to prepare students for their large exhibit designs. The remainder of the module (several days) is spent working on the major project of the module: designing a 3D art exhibit containing 40 pieces of art (from the European Renaissance) that fits within a large room located on campus. Students have the opportunity to visit the space in which the hypothetical gallery will be located with tape measures and a blank scale drawing of the room. Additionally, the room itself contains built-in constraints in the form of columns, several doors, and windows. As a result, students may add temporary walls to their exhibit, and they often consider other elements such as lighting and seating in their design. All groups construct a scale model of their gallery, which is assigned with few constraints in order to encourage creativity in students’ approaches to the problem. Instructors have received models made of cardboard, created in AutoCAD, online 3D room planners, and created in video games such as Minecraft and Fortnite. Many of the experiences built into designing an art gallery easily map onto the modeling process, and ultimately this module serves as a springboard into initial discussions on math modeling. More specifically, after defining the problem as a

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class, teams identify variables (number of people allowed in the room, spacing between artwork, how people flow through their exhibit, etc.), make assumptions related to their variables in order to make the problem more tractable, and implement their model by organizing and arranging their artwork in the room.

3.5 Leaking Pool As part of the Introduction to a Math Modeling module, students build upon their experiences with the Art Gallery module by deciding when to repair a leaking pool on campus. This problem, supplied by the athletic departments, actually occurred on campus in 2016. VMI was trying to determine if they should replace the pool in Cocke Hall, which was leaking around a rate of 10,000 gallons per night, or temporarily top it off with water every day. Students spend the majority of this activity learning how to scope out this problem, ultimately producing a mathematical model at the boards aimed at addressing if and when the pool should be replaced. Before working on the problem, students engage in a participatory modeling activity on their problem-solving process. In groups of 3, students write down their steps for their process on individual half sheets of paper. One group then tapes an individual step to the board, and other groups cluster similar steps they devised nearby. This process repeats until the entire process is on the board. The instructor discusses the order of the components of the problem-solving process and where there might be cycles within the process as a whole. From there, the instructor provides a graphic of the math modeling process and has a brief discussion about how the students’ version relates to it. Not only does this activity give students validation in their ways of thinking, but it also provides a common ground upon which different learning styles can engage with the topic of math modeling. The rest of class involves building a model for the leaky pool problem. Scaffolded questions, included below, guide students and set up a class discussion on defining the problem, variables that need consideration, and simplifying assumptions. • • • •

What information do you have? What information do you need to know? What variables will you use in your model? What assumptions do you plan to make as you scope this problem?

Based on the answers to these questions, students develop an equation or set of equations to help answer the original problem (as they defined it) about the leaky pool. While there is instructor freedom on how this is pursued, it is suggested that students build their models on the boards around the classroom. Towards the end of class, students share out their model and justifications, and the instructor facilitates discussion on the different approaches taken by students to obtain a solution.

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3.6 Army Finance The VMI Army ROTC helped develop an Army Finance module for the second semester of the course sequence. This module tasks students with building and manipulating financial calculators in Excel, exploring various retirement plans provided by the US Army, and conducting sensitivity analysis on various factors associated with these retirement plans. The ultimate deliverable for this module includes a detailed report on the findings of the sensitivity analysis along with a professional presentation of their findings. Initially, instructors provide students with guided notes designed to assist with constructing a financial calculator in Excel (i.e. for savings plans, loans, mortgages, etc.). During this time, the instructor checks in with groups, guiding students towards fixing issues with their spreadsheet tools. Afterwards, the students tackle the following problem: using the US Army’s Thrift Savings Plan, how can you achieve $30,000 by the time the rank of Captain is achieved? Many facets must be considered in approaching this problem, such as retirement fund, interest rate, personal contribution, employee matching contributions, and promotion rate. A generic spreadsheet tool is provided for groups to use (whether or not they struggled with constructing their own). This is the first major application of the modeling process students face after the Intro to Modeling module, and capitalizing on that experience is essential. Instructors encourage students to define their problem, figure out the variables in the problem, and justify any assumptions made as they solve it. Once this first task is complete, students then must analyze the impact of one of the variables in their model on their financial plan. Since this module occurs somewhat late in the semester, the instructor assigns factors based on difficulty and the strength of each group. For example, a stronger group might look at the impact of taking a fixed time period off of saving for retirement, while another group may only look at the impact of varying their rate of savings. Other variables assigned include fund type (there are 10 distinct funds in the Thrift Savings Plan), rate of return, delayed start, and time to promote. Additionally, heavy emphasis is placed on data visualization in this project. Students are challenged to look back on what they learned in the first semester as they decide how to communicate their results visually.

3.7 Other Activities We conclude this section by briefly discussing a few more example activities, most of which make up only a portion of a larger module in the Math that Matters curriculum. • The Department of History suggested that students analyze local historical populations when studying data trends and curve-fitting. As a result, students

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learn about using trendlines to interpolate and extrapolate through a case study on the population of colonial Augusta county, Virginia. • The Commandant’s staff on campus helped with several modules, such as providing fitness data for a project where students use statistics to identify the fittest company on post. They also suggested that in the unit on statistical studies, students create a metric for what it means to be a good cadet and use it to design a statistical study that investigates the impact of various behaviors on cadethood. • The Department of Economics and Business suggested that we use the CobbDouglas model for production as an example in the modeling unit, which appears again in later courses in their department. • Analyzing trends in corn syrup consumption in the USA using multiple trendlines (i.e. using a piecewise model) came from one applied mathematics faculty member’s prior work in food security. Likewise, a similar assignment looking at linear and polynomial trends in national park attendance data stemmed from another math faculty member’s work on a modeling project. Finally, the culminating experience of the course sequence includes a project where students choose a topic of interest and apply tools from the course to answer a question about that topic. Working in pairs, students spend the majority of the second semester under the supervision of their instructor applying mathematics to a meaningful problem. Projects tend to fall into one of the following categories: computing z-scores for comparing multiple statistics, using confidence intervals or hypothesis tests to draw inferences about data, creating, defending, and applying a metric, and using trendlines to analyze data. Posters are presented at a course-wide poster session at the end of the semester, and students are encouraged to invite their friends, family, and faculty from their major. These kinds of experiences, uniquely supported by an application-driven curriculum, leave students better prepared to use mathematics in the real-world.

4 Student Feedback We now use student feedback to anecdotally gauge the impact of an applied-focused curriculum. The following is a sample of quotes from two sources: (1) student evaluation surveys sent out at the end of each semester (where comments were pulled from official class evaluation reports) and (2) an in-class reflection discussion occurring at the end of a semester (with comments written by students at the board in response to questions on the students’ impressions of the usefulness of mathematics). • The most intellectually stimulating thing we did in class was learning how to compare data in practical situations. • Group activities were the most stimulating as they made you think and talk.

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• The poster project was the best part of the class because it felt like actual research, and as a non-STEM major it gave me the opportunity to use some of my skills in writing and forming arguments in a away I usually am not able to in math classes. • I really enjoyed the poster. It was one of the first times I was actually proud about something in a math class and excited to share results. It felt like it was something useful and meaningful. • This course was literally learning a marketable skill, which while not that fun was still a good learn. • This class shows applicability. It gives the practicality behind the math or a different perspective on math. • I used my skills as a history major on my thesis (analyzing populations). • This math is more tangible. • We learned that math is not just about the numbers, but about problem-solving skills. • It matters how you apply math. It is used in all fields of learning. • We can apply what we learned to real data. • We actually used practical and useful solutions to answer problems. • Problems felt smoother to get through because of the real-world relation. The comments above capture the general consensus of the students in Math that Matters. General education courses naturally reach an audience from a wide array of disciplines, and students in these courses benefit from an applied, interdisciplinary approach to mathematics. Anecdotally, students feel a stronger connection to the material, believe that mathematics is useful, and utilize skills learned in Math that Matters in other courses or projects. Furthermore, these comments suggest improved attitudes towards learning and doing mathematics, and we hope to replicate this effective response in the future. Experiences with mathematics in the real-world, successful cross-disciplinary collaborations, and the development of transferable soft-skills yield the sentiments expressed above, and applied mathematicians are the best candidates to design and teach these types of courses. Moving forward, we plan to more formally assess the impact of the course on student learning and attitudes towards mathematics. We collected data by measuring student proficiency on exam questions covering the following objectives: • Connect ideas of modern mathematics to applications in real-world settings. • Understand the relationship between variables and parameters of mathematical models and the patterns or phenomena they represent. • Formulate a problem using appropriate mathematical techniques and expressions. • Apply mathematical techniques to solve quantitative problems. • Communicate a solution in a manner that clearly indicates the line of reasoning. Students demonstrate proficiency in an objective by achieving at least 70% of the points allotted for that particular objective. In addition, we plan to analyze the effective element of the course sequence, using data obtained from surveys

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on student confidence relating to mathematical objectives in addition to questions pulled from the Attitudes Toward Mathematics Inventory [15]. Recently, we shifted to collecting attitudinal data using the MUSIC Model of Academic Motivation Inventory [16], which measures academic motivation through student perception of empowerment, usefulness, success, interest, and caring of their instructor relating to their success in the course. Ultimately, we hope to corroborate the aforementioned anecdotal evidence with strong performances in mathematical objectives alongside increased motivation and positive attitudes towards mathematics.

5 Conclusion In summary, the Math that Matters curriculum leverages the experiences of nonmathematical faculty at VMI in the creation of application-driven course content. Students are immersed in real-world problems, and focus on contextualizing the mathematics they employ in solving them. We have early anecdotal evidence of a successful implementation of the course sequence, and we believe that a multidisciplinary approach to mathematics increases student engagement, understanding of the content, and appreciation for the usefulness of the mathematics in general. However, there are many challenges associated with making a program like Math that Matters portable to other institutions. A significant portion of the course design centers on student buy-in to course materials. To that end, we incorporated elements unique to VMI in the curriculum, such as fitness data from the cadet physical training program and references to research conducted specifically by VMI faculty. While the mathematical content is transportable to other schools, the contexts and applications will depend on the non-mathematical faculty who help develop the course materials (which also depends on the students taking the course). Furthermore, the activities described in this paper involve an instructor facilitating small groups of students working together to solve problems. Adapting this style of learning at a larger institution would perhaps require the use of novel technology, large roundtable learning spaces, and/or many trained learning assistants in the classroom. The opportunity to provide students with meaningful quantitative skills contextualized by their interests is worth the bevy of challenges. Find what matters to your students, build classroom experiences that utilize skills and tools relevant to their majors and careers, and connect with faculty in other disciplines in the development of course materials. Finally, provide your students with open-ended problems steeped in context, giving equal weight to both obtaining and communicating results.

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References 1. Hughes-Hallett, D.: Achieving numeracy: The challenge of implementation. In: Mathematics and Democracy: The Case for Quantitative Literacy, pp. 93–98 (2001) 2. Kafura, D., Bart, A.C., Chowdhury, B.: Design and preliminary results from a computational thinking course. In: Proceedings of the 2015 ACM Conference on Innovation and Technology in Computer Science Education (2015) 3. National Research Council and Others.: The Mathematical Sciences in 2025. National Academies Press (2013) 4. Elrod, S.: Quantitative reasoning: The next ‘across the curriculum’ movement. Peer Review 16(3), 4 (2014) 5. Ma Zenia, A.: Developing quantitative reasoning: Will taking traditional math courses suffice? An empirical study. J. General Educ. 61(4), 305–313 (2012) 6. Abramovich, S., Grinshpan, A.Z.: Teaching mathematics to non-mathematics majors through applications. Primus 18(5), 411–428 (2008) 7. Perin, D.: Facilitating student learning through contextualization. Community College Research Center (CRCC) Working Paper, vol. 29. http://files.eric.ed.gov/fulltext/ED516783. pdf 8. Nilsson, P., Ryve, A.: Focal event, contextualization, and effective communication in the mathematics classroom. Educ. Stud. Math. 74(3), 41–258 (2010) 9. Bell, S.: Project-based learning for the 21st century: skills for the future. Clear. House J. Educ. Strateg. Issues Ideas 83(2), 39–43 (2010) 10. Stillman, G., Brown, J., Galbraith, P.: Researching applications and mathematical modelling in mathematics learning and teaching. Math. Educ. Res. J. 22(2), 1–6 (2010) 11. Bliss et al.: Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME Report). http://www.siam.org/reports/gaimme.php Cited 03 June 2016 12. Maguire Associates, Inc.: The Role of Higher Education in Career Development: Employer Perceptions. In: Chronicle of Higher Education (2012) https://chronicle.com/items/biz/pdf/ Employers%20Survey.pdf. Cited 16 Jul 2015 13. National Association of Colleges and Employers: Job Outlook 2015 (2015) https://www. naceweb.org/surveys/job-outlook.aspx. Cited 16 Jul 2015 14. Hart Research Associates: Falling Short? College Learning and Career Success. In: Association of American Colleges and Universities (2015). http://www.aacu.org/leap/public-opinionresearch/2015-survey-results. Cited 16 Jul 2015 15. Tapia, M., Marsh, G.E.: An instrument to measure mathematics attitudes. Acad. Exch. Q. 8(2), 16–22 (2004) 16. Jones, B.: Motivating students to engage in learning: The MUSIC model of academic motivation. Int. J. Teach. Learn. Higher Educ. 21(2), 272–285 (2009)

Promoting Interdisciplinary and Mathematical Modelling Through Competitions Sergey Kushnarev and Jessica M. Libertini

Abstract As we aim to prepare students to tackle increasingly complex problems in the real world, we believe that extracurricular contests can provide a fun, lowstakes environment to practice mathematical and interdisciplinary modelling skills. In this chapter, we discuss the value of providing such experiences for students and provide an overview of a popular international competition. We then describe rural and urban variations of a local modelling competition designed to provide instant feedback and foster social interaction among student modelers. After offering some thoughts on developing appropriate problems for these competitions, we close by sharing some informal feedback from students and faculty who have participated in these competitions.

1 The Value of Mathematical Modelling Experiences In many undergraduate mathematics classrooms, there is an emphasis on clearly defined problems, the application of well-documented algorithmic procedures, the discovery of theorems, and the generation of proofs. While these skills are undoubtedly important, they, by themselves, are insufficient to prepare students to tackle open-ended problems in the real world. Employer surveys consistently list problem solving, teamwork, and communication among the most important for graduates [1–4]. As discussed in [5], mathematical modelling provides authentic learning experiences that directly enhance all three of these highly sought-after skills. The value of modelling experiences has been recognized by educators, and

S. Kushnarev Singapore University of Technology and Design, Singapore, Singapore e-mail: [email protected] J. M. Libertini () Geneva Centre for Security Policy (GCSP), Geneva, Switzerland Global Studies Institute, University of Geneva, Geneva, Switzerland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 R. Buckmire, J. M. Libertini (eds.), Improving Applied Mathematics Education, SEMA SIMAI Springer Series 7, https://doi.org/10.1007/978-3-030-61717-2_5

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communities have been forming around the creation of these modelling experiences in the classroom [6]. However, in-classroom modelling experiences may artificially limit creativity; for example, students will be driven towards a differential equation for a modelling experience in a differential equations class or towards a network model in a network science class. Also, due to university schedules, in-class modelling experiences either have to be truncated to fit into a single class period or they are fragmented in time, as students have to pause their thinking to attend their other classes. Furthermore, modelling experiences in a classroom have the added gravity of assigned grades. In contrast, extracurricular modelling experiences, such as the contests discussed in this chapter, are not associated with any one course or mathematical topic, and without the fear of a poor grade, students are free to take bigger risks as they explore a broad range of possible models. While the contests are challenging, the literature shows that students who participate in these sorts of events walk away with a sense of accomplishment [7, 8], and the anecdotal evidence we present at the end of this chapter supports this conclusion.

2 A Well-Known Modelling Contest Since 1985, the Consortium for Mathematics and Its Applications (COMAP) has run its international Mathematical Contest in Modelling (MCM). In the MCM, teams of undergraduate students select one of several open-ended problems and spend a weekend developing a model or suite of models, using their models to gain insight into the topic, and writing a 20-page report showcasing their work and their findings. At the end of the twentieth century, leaders in STEM education realized a growing need for interdisciplinary thinking, and COMAP added the Interdisciplinary Contest in Modelling (ICM) as the sister competition to the MCM [9]. The competition has grown in global popularity, and in 2019, over 25,000 teams participated in the MCM/ICM contests [10].

2.1 MCM/ICM Contest Format and Problems The contest is set up as an online competition, which allows teams to participate in the comfort of their own environments without the added cost or hassle of travel. The problems are released online through multiple mirror sites. While many of the problem descriptions are too long to be included verbatim in this chapter, below are a set of topics from previous years: • • • •

Optimizing the strategic reserve of cobalt [11]; Identifying contamination in a water source [12]; Designing a safe stunt for a movie [13]; Selecting the optimal search area for a serial criminal [14];

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Determining the feasibility of a complete switch to electric vehicles [15]; Eradicating Ebola [16]; Developing a drone-based disaster relief response system [17]; Evacuating the Louvre in an attack [18].

Although there have been some minor changes in the format of the competition, the problems are all designed to give students the opportunity to apply their mathematical and interdisciplinary modelling to a problem that has stake-holders in the real world—from the stunt man who wants to survive the stunt to the personnel at the Louvre who want to optimize the evacuation of a crowd in a disaster. Currently, the MCM/ICM format provides a choice of six problems each year, and each of these problems is designed with a “low-floor, high-ceiling” so that even undergraduate students with rudimentary modelling skills should be able to make progress and produce a solution, while experienced teams have the freedom to develop as complex of a model as they feel necessary to address the problem. Therefore, the MCM/ICM sees students who participate at various stages of their undergraduate training, and regardless of their mathematical maturity, student feedback is typically very positive, citing a feeling of empowerment and accomplishment [7, 8].

2.2 MCM/ICM Judging Once the competition is over, COMAP assesses the submissions in two stages. In the triage stage, each paper receives at least two reads. The best papers from the triage phase advance to a final judging phase in which a panel convenes in person to review and discuss the final papers. Ultimately, each paper is assigned one of the following ratings: • • • • • • •

Disqualified; Unsuccessful Participant; Successful Participant; Honorable Mention; Meritorious Winners; Finalist Winners; and Outstanding Winners.

2.3 Some Drawbacks to the MCM/ICM Unfortunately, due to the size of the competition and the logistics of the judging process, the only feedback available to teams is their rating. Furthermore, the rating is only made available after the judging process is complete, and this can take several months. So, while students are excited to participate in the MCM/ICM, their enthusiasm often fades before the results are released. This means that, unless the

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team has an invested experienced coach who is willing to help them reflect on their work, the opportunity for additional growth following the competition is limited.

3 Local Modelling Competitions In [19], Galluzzo and Wendt describe a local modelling competition they designed and launched. In these competitions, a small number of teams travel to one location, work on a shorter problem for 24 h, and present their work as both a 1-page executive summary and a 10-min presentation to their peers. In 2008, the US author had the opportunity to send a team to participate in one of Galluzzo’s competitions, learned how to run one of these competitions, and began running local competitions starting in 2013. Similar to those cited by Galluzzo and Wendt, we have found several advantages to these local competitions: • Students are able to get immediate feedback on their work from both instructors and peers; • Students get a chance to see a variety of modelling approaches to the same problem; • The in-person aspect of the competition builds a sense of community, normalizes modelling behaviors, and lets students feel they are part of a larger endeavor; • While there is the added challenge of travel, there is less of a time commitment and the per-person cost is generally lower than the MCM/ICM. As the two authors held positions at universities in different geographical regions, we noticed distinct differences between our rural and urban campuses that led us to develop variations to our local competitions that responded to these differences. For example, at a rural campus, even the closest neighboring universities might be more than an hour away, while an urban school is likely to have other universities very near. So while students in an urban setting might be easily able to attend parts of the competition and travel home in between, we found that the rural competition was best run as a self-contained overnight event. Additionally, while students at a self-contained rural competition were able to dedicate themselves fully to the event, the urban competition was run over a longer period of time to account for students engaging in activities unrelated to the competition, such as jobs or family obligations. We also modified the competitions to respond to differences in modelling preparation between our two institutions. When designing a local modelling competition, we found that it was important to respond to these factors: location, culture, and prior preparation. Below, we provide further details on our rural and urban competitions in response to our local communities’ different needs.

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3.1 A Rural Competition Our rural institution was Virginia Military Institute (VMI), which is a public undergraduate-only liberal arts institution located in the scenic Shenandoah Valley of Virginia in the USA. This competition is called the Shenandoah Valley Math Modeling [sic] Challenge (SVMMC), and it has been running since 2015. Due to the remote location of this school, we had the added goal of fostering the development of relationships between students at different institutions within a few hours radius. Invitations were initially sent to faculty who had professional connections with the SVMMC directors, and pre-registration of teams was required in order to reserve the appropriate number of classrooms and meals for the event. Each year, between four and eight teams from universities in central and western Virginia and North Carolina traveled to VMI, arriving by the 1pm Saturday kick-off; two or three teams of VMI students also participated. Since the participants came from a wide range of universities, some of them had competed in similar competitions or had modelling experiences from their coursework, while others came out of curiosity and had little or no modelling experience. By traveling to the competition, there was an unstated expectation that students would largely focus on the competition during the 24 h, although students were free to budget their allotted time as they saw fit. Each team was assigned their own classroom with access to an overhead projections system. Visiting teams were informed in advance that no lodging, other than the classroom, would be provided, and that the expectation was that they could bring whatever they needed to make the classroom their own. Students often arrived with air mattresses and sleeping bags, transforming their assigned classrooms into both living and working spaces for the duration of the event. Each participating school also sent at least one faculty advisor who was responsible for securing their own lodging for one night. At the 1pm kick-off, there were short introductions and a review of the rules, following by the much-anticipated reveal of the two problems. Teams then spent the next 23 h selecting one of the two problems, developing and testing a model capable of providing insight into the topic, and using their model to analyze and make recommendations or draw conclusions in the context of the real world. Figure 1 shows a student working on their problem. During the event, the teams were not allowed to discuss the content problems with one another, but other social interaction was encouraged. The cost of a group dinner and brunch in the dining hall, snacks, a midnight pizza party, and a cake for the closing ceremony were all covered by a small registration fee paid by each team. Games and snacks were continuously available in a common area. At the end of the 23 h, students submitted a 1-page executive summary plus one additional page for figures and references. They then spent their 24th hour of the competition polishing and practicing a slide show presentation to give to their peers. Figure 2 shows students giving their presentation. Meanwhile, the faculty read and discussed the one-page summaries, assigning an MCM/ICM-style rating (Successful Participant, Honorable Mention, etc.) to be printed on the team’s

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Fig. 1 A student working on their modelling problem at the white board in his team’s classroom

Fig. 2 Students giving their presentation at VMI’s SVMMC

certificate, as well as a set of comments designed to celebrate strengths, identify areas for improvement, and encourage growth as modelers and communicators. Faculty and students also scored each presentation, resulting in a People’s Choice Award. The certificates and People’s Choice Award were presented at the closing ceremony. Additionally, each team was presented a small prize from a dollar store that was personalized to the team; for example, the team that fell asleep first might get an eye mask, while the team that was the most stealthy and rarely seen around the building might get a ninja action figure. Figure 3 shows teams at the awards

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Fig. 3 Students with their small token awards and certificates at the SVMMC award ceremony

ceremony. Following the closing ceremony, the visiting teams packed up their belongings and departed from VMI.

3.2 An Urban Competition The urban version of this mathematical modelling competition was held at Singapore University of Technology and Design (SUTD), an engineering and architecture tertiary education institution located in a densely populated city-state. Singapore is completely connected by a convenient transportation system, allowing many undergraduate students to commute to the competition site from their homes anywhere in the nation. None of the students who participated in this competition had prior experience with mathematical modelling. The SUTD Math Modelling Competition (SUTD-MMC) was organized during the term break in January, also called the Independent Term Period (IAP). SUTD maintains a popular website filled with IAP activities to help students stay engaged during the term break. The contest was prominently featured on this website, and a newsletter was printed and hung in elevators and bulletin boards around the SUTD campus. The contest director also sent emails to his former students, encouraging them to participate. While 15–20 SUTD students generally signed up each year, only 8–15 of them participated. Although this event has been limited to SUTD students, the program could easily be adapted to include multiple institutions in an urban setting. Unlike participants in the rural competition, who were physically isolated and able to spend 24 consecutive hours focused on the competition, students in the urban competition needed to fit the contest into the other aspects of their daily lives. Therefore, students are given two days to develop their solution and another half day to finalize and polish their presentation. Overall, it was expected that students in the rural and urban versions of the competition would devote a similar amount of time to addressing the problem but that this time would be distributed differently in the two settings.

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Since meals are not provided in this urban version of the competition, students were not required to pay any registration fees, although students were encouraged to register by signing up online to show their interest. Students were free to arrange themselves into teams, but many students signed up as individuals and were merged into teams for the event. Since the participating students did not have prior modelling experience, the SUTD-MMC opened with a pre-competition workshop on modelling which is described later in this chapter. The workshop was held on the Friday afternoon of the competition, and then at 10 pm, the modelling problem was sent to the participants via email. Over the next 48 h, students worked on the problem, developed a model to help them assess the situation, used their model to answer the question about the real world, and then wrote an executive summary of the most important aspects of their work and key findings. The deadline for an email submission of the executive summaries was 10 pm on Sunday. Then the teams had an additional 12 h to prepare a 10–15 min slide show presentation. Since the release of the question and the submission were both via email, the teams were free to work from any place that was convenient for them, and the host institution (SUTD) did not have to provide any space for the participants to work. Between 10 pm on Sunday and 10 am on Monday, faculty judges read the emailed submissions and prepared comments of constructive criticism to be given to the teams at the close of the in-person portion of the competition. Faculty also assigned an MCM/ICM rating (Successful Participant, Honorable Mention, etc.) to each executive summary. On Monday, all the teams emailed their slide shows and gathered in a presentation room on campus where all of the submitted files had been uploaded to the room’s projection system. After a brief welcome, each team presented their work to the collection of participants and faculty judges who rated the presentations. Figure 4 shows a student presenting his team’s work. At the end of the presentations, the students were asked to leave the room. The judges discussed each presentation, prepared specific feedback for each team. Additionally, the judges deliberated the appropriate small prizes for each team. Unlike the rural competition, where the competition director gets to know each team individually and tailors the small prize accordingly, in the urban setting, the competition director purchases several small prizes that can relate to the quality of the submission. For example, the competition director may have a stuffed fox to give the team with the most clever or creative solution, a cheap calculator or abacus toy for a team whose solution did not include enough quantitative evidence for their conclusions, and a mini-lego set for a team who had a good start but needed to keep building. The prizes were low in cost so that students did not become jealous of the prizes awarded to other teams; the value of the prizes was not in their cash value, but rather in the way the prize symbolized the pursuit of knowledge and the fun of the experience. After the deliberations, the students came back into the presentation room for a reflection session where they were encouraged to discuss their experience, such as whether things went as expected and what they did when they hit an unexpected challenge, what they learned from the presentations of the other teams, and overall

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Fig. 4 A student presentation at the SUTD-MMC

Fig. 5 Teams getting feedback and awards at the SUTD-MMC

how they felt about the modelling experience. As all of the participants were new to modelling, this reflection period was important, as it gave the students space to comment on the diversity of solutions and acknowledge that everyone was challenged and grew from the experience. Following the reflections session, the small prizes were awarded, and the faculty provided the feedback on both the presentations and the written executive summary before the competition came to an official close. Figure 5 includes photographs from the award ceremony.

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4 Problems for the Local Modelling Competition One of the challenges in running this type of competition is developing appropriately scoped problems. Unlike the MCM/ICM, these local competitions assume a much shorter time period during which students can engage with the problem. For students with less modelling experience, it may help to scaffold the problem by asking a question in three or four parts. The first part is written with strong hints nudging students towards a particular approach that will allow them to get started, gain traction, and build confidence at the start, and the subsequent parts increase in their open-endedness, giving strong and creative teams an opportunity to shine. This approach of scaffolding the questions for less experienced modelers was developed building off of the success of the MathWorks Math Modelling Challenge (M3 Challenge), a high school modelling competition run by the Society for Industrial and Applied Mathematics (SIAM) [20]. For experienced modelers and those training for the MCM/ICM, it may be helpful to remove the scaffolding so that the problem looks more like an MCM/ICM problem, but the problem scope should be smaller. Additionally, the authors work with others who run similar competitions, such as Galluzzo, to develop problems collaboratively, which simplifies this task. A good modelling problem for a weekend local competition should have three key ingredients: 1. The topic should be of interest to the participating student population, 2. There should be at least one way to answer the question relatively easily (even if badly) so that students can start with something and then focus on improving it, and 3. Any required information should be provided or relatively easy to find so that students can focus on addressing the problem instead of searching for data. Below are examples of previous problems that have been used for this type of competition. In [19], Galluzzo and Wendt share an example question from 2010 in which students consider the future of the US Postal Service through a financial lens. While this seems like a large problem, even beginning modelling students were able to estimate the revenue by looking up postal rates as well as historical trends on mail volume. They can estimate operating costs by looking at employee salaries and benefits, and they can continue to refine their model by including additional complexities. In 2015, one of the SVMMC questions had students develop a ranking system for universities to prioritize access to financial aid. This problem, like many ranking or metric-development problems, fits nicely into almost any time frame as it is relatively easy to come up with a rudimentary metric as a starting point. Students then spend most of the competition refining their metric based on their mathematical skill level. Students with less mathematical skills might simply work to scale and combine factors appropriately, or explore how adding or removing factors impacts their results as they work to validate their model. Students with more advanced skills

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might build a training set to create a classification algorithm, identify factors, and/or select appropriate weights for those factors. For the 2019 SUTD-MMC, we developed a problem (jointly with two faculty members from the Architecture Department) asking to model when it is better to refurbish an older public housing apartment building, or to demolish and build a new one. For the context, 85% of Singapore population lives in public housing apartments. Other problems in the SVMMC and SUTD-MMC have covered topics such as finding an equitable approach for distributing resettling Syrian refugees across the European nations and developing an evacuation plan for a national park during a wildfire.

5 Possible Modifications to Customize the Event While the competition is a worthwhile event on its own, there are benefits to offering complementary activities such as a social event or workshops. Below we describe a few possible workshops, including one that we have run for the SUTD students.

5.1 Introductory Modelling Workshop for Students Students at SUTD participate in their local competition prior to having curricular exposure to mathematical modelling. Therefore, we felt it was important to run a workshop introducing the modelling process and giving students the opportunity to solve a problem together before starting the competition. Since the SUTD-MMC officially started with the release of the questions on a Friday night, a two to three hour modelling workshop was scheduled for Friday afternoon. It was at this workshop that individual registrants were put into teams for both the training and the competition. At the start of the workshop, the students were given a chance to form teams of three to five students, depending on the number of registrants. The competition organizer then went over the structure of the competition, the requirements, and the rules. Then we showed an overhead slide showing the steps of the mathematical modelling process—from the formulation of a question based on the real-world issue and the identification of factors to the development of the model used to get a solution and the communication of the findings. Armed with this overview, we then posed a modelling problem to students; for example, last year, the practice question was to examine the effect of climate change on Singapore. Then, students were given about 15 min to brainstorm. As they worked to frame the problem, they developed lists of factors that might be relevant—things like current and predicted levels of sea rise and elevation charts for the small island nation. The teams shared their thoughts. This sort of experience can be very

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emotionally rewarding for students in multiple ways. First, the validity of their work is affirmed for ideas that are echoed by other groups. Second, their creativity is recognized for any response that is uniquely theirs. And third, their curiosity is engaged by new ideas contributed by others. Next, teams are given some time to develop a plan of attack for their analysis. In this stage, teams come up with one or more ideas for possible models, do research to see which of their plans is most viable based on available data, and ultimately select a course of action. At this point, students are making modelling choices about their approach: stochastic or deterministic, first principles or empirical, the coding platform, etc. Once they have had a chance to develop a plan of attack but have not had time to fully implement it, groups once again share their ideas with everyone at the workshop. This gives everyone a chance to explicitly note the diverse set of approaches and helps dispel the common misconception that mathematics problems can only be correctly solved with one solution technique. This also gives everyone the chance to offer and receive feedback on their approaches, and it leads into a nice discussion about model assessment, including the inherent strengths and limitations of a given approach. Figure 6 shows teams hard at work considering possible sources and approaches for their work. After a discussion of model assessment, students are guided through a discussion about assumptions, talking about those used to initially make the problem tractable, those that are inherent in the model choice, and those that help tie the model to the real world. Towards the end of the workshop, the organizer discusses the later steps in the modelling process—how, after implementing their model, they need to validate it and that this may lead to refinements of the model, and how, after getting a solution, the most important aspects of the problem, approach, and findings can be captured effectively in a well-written executive summary. After time for student questions and open discussion, the students are released from the workshop to await the official start of the competition.

Fig. 6 Students hard at work during the SUTD-MMC introduction to modelling workshop

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5.2 Other Possible Workshops In addition to the student workshop as described, there are opportunities to run other workshops. Modelling workshops for students can also focus on specific aspects of the modelling process where students traditionally face challenges, such as scoping the problem, making and justifying assumptions, assessing the model, and producing good expository writing about their work that effectively targets the appropriate audience. One activity that has been done at VMI is a short workshop on writing executive summaries. This workshop is run in our capstone course, which is taken by senior applied mathematics majors. In the workshop, a problem, model, and solution to a problem are all given to them, and they spend 30 min writing an executive summary. We then bring up one of the summaries on the project, and as a class, we edit it line by line—looking for ways to tighten the writing, making sure that the points are clearly made, and checking that the summary includes sufficient information about the problem, the methodology, and the findings. In addition to offering workshops for students, these competitions are opportunities to provide professional development for faculty. For example, at SUTD, one of the goals of involving the faculty in the judging of the competition is to train faculty to teach mathematical modelling and to assess student work. Many faculty experience a challenging cultural shift as they migrate from grading problems that have an obviously correct solution, such as a calculus problem, to grading modelling problems where students have significantly more autonomy and the solution space is uncharted territory [5, 21, 22]. The competition provides a low-stakes (since no student grades are impacted) opportunity for faculty to read the same papers and collectively discuss the strengths and weaknesses seen by each individual instructor. SIMIODE, which runs a similar competition focused on differential equations, has offered opportunities for faculty to learn about implementing modelling experiences in the classroom [23].

6 Student and Faculty Feedback While we have not conducted a formal study, we have collected free-response surveys from students who participated in the SUTD-MMC competition. This paper was the result of early explorations, and in the future, we plan to survey students at both competitions. We have also collected informal feedback from faculty advisors from the SUTD-MMC and the SVMMC.

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6.1 Student Feedback Overall, students are glad they participated in the event. We have organized student input around their motivation (why they wanted to participate), expectations (what they expected), their take-aways (what they got out of participating).

6.1.1

Student Motivation: Why Do You Want to Attend the Competition?

This question was asked of students as part of the registration process in the SUTDMMC. This gives us a pre-competition perspective of why students wanted to participate. Out of thirty registrants who provided an answer to this question, the majority of the answers fell into one of three categories: they cite wanting to learn more about mathematics or modelling, they cite believing the event will be interesting or fun, or they cite wanting to experience the competition setting.

Learning More Twelve of the thirty responses directly talked about wanting to learn or practice their mathematics or mathematical modelling skills. These students specifically talked about challenging themselves, learning new things (such as mathematical modelling), hone existing skills, and apply their mathematical knowledge to real problems. Below are several examples of real student input. learn math and try to apply knowledge to solve real problems I want to learn about math modelling I want to learn more skills and gain more knowledge about maths modelling. practice my modelling skills For learning/apply things I’ve learnt Learn more Challenge my self [sic] and learn some new things.

Fun and Interesting One third of the responses, ten out of thirty, explicitly stated that the competition sounded interesting or fun, or that the respondent liked mathematics. Given that this event was fully optional and run during a break between semesters, it is not surprising that it attracted many students who participated because it sounded fun to them. Below are several examples of real student input.

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Do it for fun. IT LOOKS COOL. Math is fun Sounds interesting

Practicing for Future Competitions It is clear that several students were hoping to gain experiences that would help them in future mathematical modelling competitions, such as COMAP’s MCM/ICM. Four of the responses reflected this explicitly; these responses are listed below. Practice for ICM To practice for MCM To learn modelling and to have competition experience I would like to get a feel for math modelling competitions

6.1.2

Student Expectation: What Do You Expect from This Competition?

Similarly, during the registration process for the SUTD-MMC, students were asked about their expectations for the competition. These answers generally fell into several categories: learning new things or practicing (skills, modelling, strategies, teamwork); having fun, being challenged, or doing something interesting; and preparing for future events.

Learning and Practicing Out of the twenty-nine recorded responses to this question, nineteen directly talk about their expectation to learn, improve, or do mathematics and/or modelling in the competition. While some students expected to learn something new, others saw this as an opportunity to further develop their current skills. Below are some examples of real student responses. The basic foundation of math modelling. learning math and teamwork Learn how to model things Learn new tricks, see what kind of problems have been modelled Novel modelling techniques A good experience with like-minded people, and maybe more knowledge on useful strategies for mathematical modelling

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We call specific attention to the last item of the list in which the student says they want to learn more about the different approaches people take and the common mistakes that people make in solving modelling problems. This response is important as it highlights two things. In the first part of their response, this student acknowledges the value of seeing multiple approaches to the same problem— something that is not a typical part of the MCM/ICM but is an integral part of these local competitions. The second part of this student’s response is extremely noteworthy, especially in the context of Singaporean culture where mistakes are often seen as shameful. Educational theory talks about the idea of productive failure, but this is difficult to address in a culture like the one in Singapore [24, 25]. By removing this sort of exercise from the graded environment of a classroom and holding this introduction to modelling as an extracurricular event, the stigma of failure was removed for this student, allowing for the value of productive failure to be recognized and celebrated as an opportunity for all to learn.

Challenging, Interesting, and Fun Four of the student responses explicitly focused on their expectations for the event to be challenging, interesting or fun. All four of these responses are below. Fun and interesting problems Fun challenging, exciting questions An interesting and challenging topic to work on

Preparing for the Future Some students explicitly shared that they expected the competition to prepare them for future events, including coursework and future competitions. Below are two examples of student comments highlighting their expectation of the event to prepare them for a future course or competition. We hope to be able to at the same time gain other useful skills that can be used to investigate models, and can potentially be useful in the course of our ESD projects [Engineering Systems and Design department at SUTD] and capstone projects as well. Practice on a problem before going for MCM/ICM.

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Student Reflections After the Competition

Immediately following the competition, students participated in a reflection session. However, in the interest of promoting candid discussion, comments made there were not recorded. Later, participants in the SUTD-MMC were asked, via email, to share their reflections on the event, and it is these email responses that are used in this paper. Since this request was not part of the registration process and not required, only a limited number of students responded. Therefore, we cannot draw inferences from this population about the overall student perspective; rather we consider these observations as individual examples of perception. Below we talk about the feedback provided in several of these submissions.

The Competition and Introductory Workshop One student wrote about how the workshop prepared him for the competition and how the competition both met his expectations and surprised him. This student also commented on the shorted time frame, noting it as a positive. Below is an excerpt from this student. The competition is very well suited for students that are interested in learning about math modelling. The introductory lessons provided are more than sufficient in getting students into the spirit of math modelling, and also gets team members (often strangers) warmed up to each other. The short time frame also encourages more participants to join, and helps participants focus. I feel that the format of presentations is appropriate as it is able to fully display the thought processes of each individual participant, and highlights the contributions of each participant. Going into the competition, I generally expected to be able to work on an interesting project while gaining more experience with math modelling and the thought processes behind it. Unexpectedly, through the competition, I also gained a lot from getting feedback about our team’s presentation, and listening to other teams’ ideas. The judges and instructors were also very open-minded and supportive with regards to different approaches and justifications.

Other students reflected more on the value of mathematical modelling and their experience of working through the modelling process. Below are three excerpts from students who reflected on this aspect of the experience. Mathematics can help to address many of the big problems in the world. You make assumptions to simplify the problem into logical statements. Then you experiment with models to find some that suitable for the problem. You may even make predictions and estimate its accuracy. You also analyze the model by identifying which parameters are important and how they affect your solution. Your solution is usually not complete, but you have a solution that is backed with data and logic. Mathematical modelling helps you to identify the important problems to tackle. Some examples that I have tried with my friends—Is making all buildings green better for the environment? What is the true value of your data? Who should we save first when there is a forest fire? Does a dragon waste more energy flying or breathing fire?

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S. Kushnarev and J. M. Libertini We learn mathematics to solve problems, and here we apply them in action. After the initial research, your team will need to agree on the approach to the problem, the work for each of the members based on their ability. Your team will commit to the plan while expecting changes depending on further results. You will learn about problem solving, collaboration and presentation skills.

6.2 Faculty Feedback In addition to gathering student perceptions, we wanted to understand the perspective of the faculty involved in both the SVMMC and the SUTD-MMC.

6.2.1

SVMMC Faculty Feedback

The faculty advisors who attended the event included both pre-tenure and tenured faculty. While some had experience teaching modelling, others were pure mathematicians. When asked to comment about the SVMMC, common themes emerged, including a focus on the social nature of the competition and the opportunity to tackle a real-world problem. One faculty member provided the following set of comments. The SVMMC is a wonderful and intimate experience for students. It provides a sense of camaraderie with students from other colleges and universities by sharing meals and solution ideas at the end. Seeing how other students approached the same problem, and explain it, is a valuable experience in itself. It also provides same day feedback for students on their solutions with guidance from experienced faculty at other universities who sit down with them and give them constructive and encouraging guidance. The whole experience has a great community feel for both the students and faculty. I will also say that the faculty who run the competition have been especially aware of the goal being academic and community development rather than a more competitive and exclusionary event. They students are made welcome at every moment and the clever and playful awards at the end bring a very positive feel to the event for every participant. This event takes a long view of academic development. Its success should not be gauged by the success of a winning team, but the persistence of all participants later in their careers in academic fields involving creative problem solving and mathematical modelling. It is a great experience.

In an article in her university’s newspaper [26], another faculty member provides the following quote. There was a real value to having students present their results, see other group’s solutions, and get immediate feedback. Our students found it interesting to see how other people had attacked the same problem. They also noticed that the students from VMI had more of a military mindset, which made them solve the problem from a very different perspective.

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SUTD-MMC Faculty Feedback

The faculty who were involved as facilitators and judges had several different position titles including faculty fellow, lecturer, and assistant professor. In addition to facilitators and judges, the problem author, who is a member of the faculty in a different department, was invited to share his perspective.

Affective Outcomes A common theme in faculty reactions is the way in which the competition, and specifically the modelling process, allowed students to view mathematics more favorably, as modelling focuses on creativity and real problems. Many faculty comments either fully focused on the affective element of the contest or drew a connection between the affective and cognitive goals of modelling. Below are some excerpts from faculty to this effect. It’s a great opportunity for students to apply math creatively to solve real life problems and also to have fun with math. Students appreciate the usefulness of math through math modelling compared to typical textbook problems. In order to solve the modelling problems, students have to work in group, and learn how to formulate problems, make assumptions, search for resources and analyze solutions, which are some valuable skills in both academia and industry. Students get to see the practical relevance of mathematics in real life applications, and hopefully would inspire them to take mathematics seriously at university. The math modelling is an [sic] very interesting and highly useful activity, to promote the “applications” of math in everyday life. The applications of math are probably the most serious lack in our current math education. As a result, many students learn, forget and dislike math, and find it useless. Through the math modelling competition, I see the excitement from students, and feel truly glad that they can use model and data to provide constructive advice on various issues. Their performances also gave me inspirations to improve my teaching of math, to make it more relevant and accessible.

The Workshop One professor specifically focused his comments on the workshop noting that students arrived without any background but left ready to be successful in the competition. The specific comments from that professor include the following excerpt. At the beginning of the orientation day, the students were unsure on how to approach things but spirits remained high. . . . The students left the orientation with a much better idea on how to tackle the math modelling task that they need to work over the weekend.

Lastly, we felt it is important to share the feedback provided by the author of the 2019 competition problem who is an assistant professor in a different

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department. While his full feedback is provided below, we call attention to his positive impressions of the variety of approaches. Earlier this year, we had proposed a topic for the Math Modelling competition on resource consumption in residential buildings. We asked for solutions to estimate the impact of embodied energy (all resources in the construction process) in comparison to operational energy (all resources during the lifespan of a building). From many appliances, we hear that it is better to replace old models every few years as newer technological developments would help reduce the environmental impact. In regard to buildings, we were curious if it applies in a similar way, or if buildings should be retained as long as possible due to the amount of resources needed to make them. The participants found very interesting and creative ways to collect data and to make sense of it. Their approaches ranged from breaking down buildings into different phases in their life cycles in great detail, to looking at the larger context, taking electricity consumption per residential areas and relating it to the average completion dates. In all cases, participants produced well structured, very detailed and methodological reports. The results proved very useful as they represented different possible perspectives and opened up unexpected views on the challenge.

7 Conclusions In conclusion, local competitions provide a unique space for students to engage with mathematical modelling. As with larger competitions, such as MCM/ICM, students have the opportunity to work on an open-ended challenging problem that requires them to take ownership of their decisions throughout the modelling process. Unlike the MCM/ICM, these local competitions provide an opportunity for students to learn from the diversity of approaches taken by other teams and gain immediate feedback on their own work. Since this competition takes place outside of the classroom, students are more open to trying new things and learning from productive failure, which is particularly important in cultures were failure is considered shameful. In conclusion, local competitions provide a unique space for students to engage with mathematical modelling. These local competitions are low-budget events that allow students to learn about modelling, develop their skills, and learn from the successes and mistakes of others, in a fun and supportive environment that is not tied to a grade in a class or the stress of an international competition.

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